| L(s) = 1 | − 11·3-s + 25·5-s − 35·7-s + 27·9-s + 9·11-s + 91·13-s − 275·15-s + 161·17-s − 84·19-s + 385·21-s + 60·23-s + 375·25-s + 108·27-s + 217·29-s + 22·31-s − 99·33-s − 875·35-s + 384·37-s − 1.00e3·39-s + 182·41-s + 396·43-s + 675·45-s + 61·47-s + 735·49-s − 1.77e3·51-s + 448·53-s + 225·55-s + ⋯ |
| L(s) = 1 | − 2.11·3-s + 2.23·5-s − 1.88·7-s + 9-s + 0.246·11-s + 1.94·13-s − 4.73·15-s + 2.29·17-s − 1.01·19-s + 4.00·21-s + 0.543·23-s + 3·25-s + 0.769·27-s + 1.38·29-s + 0.127·31-s − 0.522·33-s − 4.22·35-s + 1.70·37-s − 4.10·39-s + 0.693·41-s + 1.40·43-s + 2.23·45-s + 0.189·47-s + 15/7·49-s − 4.86·51-s + 1.16·53-s + 0.551·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 5^{5} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{25} \cdot 5^{5} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.348258004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.348258004\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{5} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 11 T + 94 T^{2} + 629 T^{3} + 3397 T^{4} + 18728 T^{5} + 3397 p^{3} T^{6} + 629 p^{6} T^{7} + 94 p^{9} T^{8} + 11 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 9 T + 2238 T^{2} - 23087 T^{3} + 4167445 T^{4} - 21189816 T^{5} + 4167445 p^{3} T^{6} - 23087 p^{6} T^{7} + 2238 p^{9} T^{8} - 9 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 7 p T + 5492 T^{2} - 409177 T^{3} + 24386719 T^{4} - 1100205720 T^{5} + 24386719 p^{3} T^{6} - 409177 p^{6} T^{7} + 5492 p^{9} T^{8} - 7 p^{13} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 161 T + 23708 T^{2} - 2544387 T^{3} + 237642227 T^{4} - 17410042744 T^{5} + 237642227 p^{3} T^{6} - 2544387 p^{6} T^{7} + 23708 p^{9} T^{8} - 161 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 84 T + 28807 T^{2} + 1691344 T^{3} + 341232202 T^{4} + 15169698424 T^{5} + 341232202 p^{3} T^{6} + 1691344 p^{6} T^{7} + 28807 p^{9} T^{8} + 84 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 60 T + 26899 T^{2} - 982896 T^{3} + 373604122 T^{4} - 12179421480 T^{5} + 373604122 p^{3} T^{6} - 982896 p^{6} T^{7} + 26899 p^{9} T^{8} - 60 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 217 T + 116164 T^{2} - 19414395 T^{3} + 5566959143 T^{4} - 689049337336 T^{5} + 5566959143 p^{3} T^{6} - 19414395 p^{6} T^{7} + 116164 p^{9} T^{8} - 217 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 22 T + 86135 T^{2} + 510448 T^{3} + 3890663030 T^{4} + 56768173900 T^{5} + 3890663030 p^{3} T^{6} + 510448 p^{6} T^{7} + 86135 p^{9} T^{8} - 22 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 384 T + 209885 T^{2} - 62242688 T^{3} + 18710821870 T^{4} - 4337259308416 T^{5} + 18710821870 p^{3} T^{6} - 62242688 p^{6} T^{7} + 209885 p^{9} T^{8} - 384 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 182 T + 6221 p T^{2} - 56531208 T^{3} + 28927386338 T^{4} - 6061263529412 T^{5} + 28927386338 p^{3} T^{6} - 56531208 p^{6} T^{7} + 6221 p^{10} T^{8} - 182 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 396 T + 286879 T^{2} - 75727600 T^{3} + 33051211354 T^{4} - 7104558938824 T^{5} + 33051211354 p^{3} T^{6} - 75727600 p^{6} T^{7} + 286879 p^{9} T^{8} - 396 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 61 T + 134078 T^{2} + 9472629 T^{3} + 22937879153 T^{4} - 2040087074768 T^{5} + 22937879153 p^{3} T^{6} + 9472629 p^{6} T^{7} + 134078 p^{9} T^{8} - 61 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 448 T + 578557 T^{2} - 219016384 T^{3} + 152832703422 T^{4} - 46579745917952 T^{5} + 152832703422 p^{3} T^{6} - 219016384 p^{6} T^{7} + 578557 p^{9} T^{8} - 448 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 + 132 T + 870271 T^{2} + 84985584 T^{3} + 326565073162 T^{4} + 23961484405848 T^{5} + 326565073162 p^{3} T^{6} + 84985584 p^{6} T^{7} + 870271 p^{9} T^{8} + 132 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 602 T + 655377 T^{2} - 362351704 T^{3} + 255658990130 T^{4} - 100356900567804 T^{5} + 255658990130 p^{3} T^{6} - 362351704 p^{6} T^{7} + 655377 p^{9} T^{8} - 602 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 332 T + 1351431 T^{2} - 345465872 T^{3} + 765254730298 T^{4} - 147725985096840 T^{5} + 765254730298 p^{3} T^{6} - 345465872 p^{6} T^{7} + 1351431 p^{9} T^{8} - 332 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 464 T + 1248403 T^{2} + 532544704 T^{3} + 776085622858 T^{4} + 258895536250080 T^{5} + 776085622858 p^{3} T^{6} + 532544704 p^{6} T^{7} + 1248403 p^{9} T^{8} + 464 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 1206 T + 2301349 T^{2} - 1890555208 T^{3} + 1925960429554 T^{4} - 1105711685256004 T^{5} + 1925960429554 p^{3} T^{6} - 1890555208 p^{6} T^{7} + 2301349 p^{9} T^{8} - 1206 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 49 T + 1836402 T^{2} - 68363625 T^{3} + 1508137355601 T^{4} - 89026735010928 T^{5} + 1508137355601 p^{3} T^{6} - 68363625 p^{6} T^{7} + 1836402 p^{9} T^{8} + 49 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 20 T + 1629367 T^{2} + 251465072 T^{3} + 17140038174 p T^{4} + 235905370512568 T^{5} + 17140038174 p^{4} T^{6} + 251465072 p^{6} T^{7} + 1629367 p^{9} T^{8} + 20 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 1634 T + 3317269 T^{2} - 3344265592 T^{3} + 4163102582578 T^{4} - 3146161683913356 T^{5} + 4163102582578 p^{3} T^{6} - 3344265592 p^{6} T^{7} + 3317269 p^{9} T^{8} - 1634 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 1737 T + 3314636 T^{2} - 3865987819 T^{3} + 4980092538739 T^{4} - 4778039753429528 T^{5} + 4980092538739 p^{3} T^{6} - 3865987819 p^{6} T^{7} + 3314636 p^{9} T^{8} - 1737 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.61850290698021073457412513015, −5.49500287449259547433309073146, −5.21429579752279234422078867854, −5.18202699942552594435116020741, −5.17180562236781172049664290269, −4.62742161661247971534063171378, −4.38002070410846075988037959288, −4.20181854616023223685932812490, −4.11584649635998034630658380571, −3.93277202774889618149489568229, −3.47982545416781974361446423376, −3.24543605496320051991670670419, −3.06751444599004621198537617939, −3.05557826671755710515279449736, −2.98054794978590934672932787416, −2.37989360297536099305257008959, −2.16290725137243830139623569556, −2.01141708924242008097220223330, −1.98342469226202122632654624858, −1.33657733298652250746620443017, −0.979830482889201669029856083448, −0.905081591657202615581037220077, −0.73912593280595603400899401937, −0.67641547745460717394280344094, −0.32584377358104612954104689038,
0.32584377358104612954104689038, 0.67641547745460717394280344094, 0.73912593280595603400899401937, 0.905081591657202615581037220077, 0.979830482889201669029856083448, 1.33657733298652250746620443017, 1.98342469226202122632654624858, 2.01141708924242008097220223330, 2.16290725137243830139623569556, 2.37989360297536099305257008959, 2.98054794978590934672932787416, 3.05557826671755710515279449736, 3.06751444599004621198537617939, 3.24543605496320051991670670419, 3.47982545416781974361446423376, 3.93277202774889618149489568229, 4.11584649635998034630658380571, 4.20181854616023223685932812490, 4.38002070410846075988037959288, 4.62742161661247971534063171378, 5.17180562236781172049664290269, 5.18202699942552594435116020741, 5.21429579752279234422078867854, 5.49500287449259547433309073146, 5.61850290698021073457412513015
