| L(s) = 1 | − 15·3-s − 25·5-s − 7·7-s + 135·9-s + 64·11-s + 80·13-s + 375·15-s − 21·17-s − 52·19-s + 105·21-s + 115·23-s + 375·25-s − 945·27-s − 277·29-s − 289·31-s − 960·33-s + 175·35-s − 303·37-s − 1.20e3·39-s − 95·41-s − 372·43-s − 3.37e3·45-s − 462·47-s − 969·49-s + 315·51-s − 73·53-s − 1.60e3·55-s + ⋯ |
| L(s) = 1 | − 2.88·3-s − 2.23·5-s − 0.377·7-s + 5·9-s + 1.75·11-s + 1.70·13-s + 6.45·15-s − 0.299·17-s − 0.627·19-s + 1.09·21-s + 1.04·23-s + 3·25-s − 6.73·27-s − 1.77·29-s − 1.67·31-s − 5.06·33-s + 0.845·35-s − 1.34·37-s − 4.92·39-s − 0.361·41-s − 1.31·43-s − 11.1·45-s − 1.43·47-s − 2.82·49-s + 0.864·51-s − 0.189·53-s − 3.92·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{5} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{5} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7524664582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7524664582\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{5} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{5} \) |
| 23 | $C_1$ | \( ( 1 - p T )^{5} \) |
| good | 7 | $C_2 \wr S_5$ | \( 1 + p T + 1018 T^{2} + 9249 T^{3} + 514729 T^{4} + 4507840 T^{5} + 514729 p^{3} T^{6} + 9249 p^{6} T^{7} + 1018 p^{9} T^{8} + p^{13} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 64 T + 3681 T^{2} - 117112 T^{3} + 5668204 T^{4} - 184122704 T^{5} + 5668204 p^{3} T^{6} - 117112 p^{6} T^{7} + 3681 p^{9} T^{8} - 64 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 80 T + 7315 T^{2} - 516336 T^{3} + 31819672 T^{4} - 1431963776 T^{5} + 31819672 p^{3} T^{6} - 516336 p^{6} T^{7} + 7315 p^{9} T^{8} - 80 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 21 T + 3166 T^{2} - 409395 T^{3} + 41452921 T^{4} + 917573292 T^{5} + 41452921 p^{3} T^{6} - 409395 p^{6} T^{7} + 3166 p^{9} T^{8} + 21 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 52 T + 25221 T^{2} + 1144472 T^{3} + 295776776 T^{4} + 10720605672 T^{5} + 295776776 p^{3} T^{6} + 1144472 p^{6} T^{7} + 25221 p^{9} T^{8} + 52 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 + 277 T + 100750 T^{2} + 16261193 T^{3} + 3470889241 T^{4} + 439743969324 T^{5} + 3470889241 p^{3} T^{6} + 16261193 p^{6} T^{7} + 100750 p^{9} T^{8} + 277 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 289 T + 142446 T^{2} + 32795447 T^{3} + 8309846369 T^{4} + 1442693508816 T^{5} + 8309846369 p^{3} T^{6} + 32795447 p^{6} T^{7} + 142446 p^{9} T^{8} + 289 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 303 T + 104684 T^{2} + 7056469 T^{3} - 992543561 T^{4} - 923313574648 T^{5} - 992543561 p^{3} T^{6} + 7056469 p^{6} T^{7} + 104684 p^{9} T^{8} + 303 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 95 T + 99962 T^{2} + 22725295 T^{3} + 11079999697 T^{4} + 1701340874364 T^{5} + 11079999697 p^{3} T^{6} + 22725295 p^{6} T^{7} + 99962 p^{9} T^{8} + 95 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 + 372 T + 370131 T^{2} + 103756368 T^{3} + 56832864686 T^{4} + 11936686771896 T^{5} + 56832864686 p^{3} T^{6} + 103756368 p^{6} T^{7} + 370131 p^{9} T^{8} + 372 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 462 T + 407497 T^{2} + 134313100 T^{3} + 76328209072 T^{4} + 19652471417164 T^{5} + 76328209072 p^{3} T^{6} + 134313100 p^{6} T^{7} + 407497 p^{9} T^{8} + 462 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 73 T + 658194 T^{2} + 37344517 T^{3} + 183327280657 T^{4} + 7893436099556 T^{5} + 183327280657 p^{3} T^{6} + 37344517 p^{6} T^{7} + 658194 p^{9} T^{8} + 73 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 529 T + 759472 T^{2} - 338138165 T^{3} + 248772820159 T^{4} - 94091561674092 T^{5} + 248772820159 p^{3} T^{6} - 338138165 p^{6} T^{7} + 759472 p^{9} T^{8} - 529 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 174 T + 902655 T^{2} - 154803156 T^{3} + 368992472876 T^{4} - 51249182518908 T^{5} + 368992472876 p^{3} T^{6} - 154803156 p^{6} T^{7} + 902655 p^{9} T^{8} - 174 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 + 919 T + 937554 T^{2} + 475015829 T^{3} + 296633796785 T^{4} + 121206972479088 T^{5} + 296633796785 p^{3} T^{6} + 475015829 p^{6} T^{7} + 937554 p^{9} T^{8} + 919 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 141 T + 549124 T^{2} + 52290413 T^{3} + 238121821471 T^{4} - 34456431877732 T^{5} + 238121821471 p^{3} T^{6} + 52290413 p^{6} T^{7} + 549124 p^{9} T^{8} + 141 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 448 T + 1181919 T^{2} - 566999424 T^{3} + 662413820568 T^{4} - 307997913886368 T^{5} + 662413820568 p^{3} T^{6} - 566999424 p^{6} T^{7} + 1181919 p^{9} T^{8} - 448 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 496 T + 2468923 T^{2} + 961456320 T^{3} + 2435790600346 T^{4} + 706932391909024 T^{5} + 2435790600346 p^{3} T^{6} + 961456320 p^{6} T^{7} + 2468923 p^{9} T^{8} + 496 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 1219 T + 2880084 T^{2} - 2362785019 T^{3} + 3149538474667 T^{4} - 1889333855618684 T^{5} + 3149538474667 p^{3} T^{6} - 2362785019 p^{6} T^{7} + 2880084 p^{9} T^{8} - 1219 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 1492 T + 3115529 T^{2} - 3869923760 T^{3} + 4084697078014 T^{4} - 3978146269760568 T^{5} + 4084697078014 p^{3} T^{6} - 3869923760 p^{6} T^{7} + 3115529 p^{9} T^{8} - 1492 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 574 T + 3927753 T^{2} - 1626463184 T^{3} + 6513965276054 T^{4} - 2009277806386788 T^{5} + 6513965276054 p^{3} T^{6} - 1626463184 p^{6} T^{7} + 3927753 p^{9} T^{8} - 574 p^{12} T^{9} + p^{15} T^{10} \) |
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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
−5.14883734128364077080955277837, −5.08507983663948270306416486277, −5.00627415273067001108919420029, −4.95958870183269163620583133261, −4.90172657750848091154394944285, −4.27054720605647958274151052821, −4.16966901964803224488321262121, −4.14642080974425248438742613566, −4.03303881070261661231421675094, −3.93528757236961645969263372137, −3.34962141895853720659039763585, −3.28005234798654344294976684607, −3.27908313360277239924360217885, −3.23478314426151298737099291495, −3.00625442678589447571309203812, −2.05110999813603549331097820231, −1.86102189700654596941681622546, −1.74972058605997499071784166022, −1.68291167539559932001286202717, −1.63434602044310504864892806750, −0.846096578006152281453282543995, −0.75337786007973999462138196838, −0.66788316937469964229433947967, −0.41414024380916888802996829574, −0.19067417091057309131433568454,
0.19067417091057309131433568454, 0.41414024380916888802996829574, 0.66788316937469964229433947967, 0.75337786007973999462138196838, 0.846096578006152281453282543995, 1.63434602044310504864892806750, 1.68291167539559932001286202717, 1.74972058605997499071784166022, 1.86102189700654596941681622546, 2.05110999813603549331097820231, 3.00625442678589447571309203812, 3.23478314426151298737099291495, 3.27908313360277239924360217885, 3.28005234798654344294976684607, 3.34962141895853720659039763585, 3.93528757236961645969263372137, 4.03303881070261661231421675094, 4.14642080974425248438742613566, 4.16966901964803224488321262121, 4.27054720605647958274151052821, 4.90172657750848091154394944285, 4.95958870183269163620583133261, 5.00627415273067001108919420029, 5.08507983663948270306416486277, 5.14883734128364077080955277837
