| L(s) = 1 | − 15·5-s + 9·7-s − 18·11-s − 21·13-s − 84·17-s + 21·19-s − 48·23-s + 150·25-s + 36·29-s + 324·31-s − 135·35-s + 33·37-s − 114·41-s + 282·43-s − 282·47-s − 360·49-s − 222·53-s + 270·55-s + 276·59-s + 303·61-s + 315·65-s + 1.03e3·67-s − 510·71-s + 447·73-s − 162·77-s + 777·79-s − 78·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.485·7-s − 0.493·11-s − 0.448·13-s − 1.19·17-s + 0.253·19-s − 0.435·23-s + 6/5·25-s + 0.230·29-s + 1.87·31-s − 0.651·35-s + 0.146·37-s − 0.434·41-s + 1.00·43-s − 0.875·47-s − 1.04·49-s − 0.575·53-s + 0.661·55-s + 0.609·59-s + 0.635·61-s + 0.601·65-s + 1.88·67-s − 0.852·71-s + 0.716·73-s − 0.239·77-s + 1.10·79-s − 0.103·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.591902811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.591902811\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 9 T + 9 p^{2} T^{2} - 650 T^{3} + 9 p^{5} T^{4} - 9 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 18 T + 912 T^{2} - 28510 T^{3} + 912 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 21 T + 2910 T^{2} + 4721 p T^{3} + 2910 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 84 T + 14706 T^{2} + 738652 T^{3} + 14706 p^{3} T^{4} + 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 21 T + 15189 T^{2} - 149614 T^{3} + 15189 p^{3} T^{4} - 21 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 48 T + 36654 T^{2} + 1165994 T^{3} + 36654 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 36 T + 31512 T^{2} + 1278578 T^{3} + 31512 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 324 T + 70560 T^{2} - 10091392 T^{3} + 70560 p^{3} T^{4} - 324 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 33 T + 107211 T^{2} - 491142 T^{3} + 107211 p^{3} T^{4} - 33 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 114 T + 125619 T^{2} + 13866756 T^{3} + 125619 p^{3} T^{4} + 114 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 282 T + 220578 T^{2} - 43728712 T^{3} + 220578 p^{3} T^{4} - 282 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 6 p T + 152826 T^{2} + 69794468 T^{3} + 152826 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 222 T + 282615 T^{2} + 72010284 T^{3} + 282615 p^{3} T^{4} + 222 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 - 303 T + 620067 T^{2} - 124220266 T^{3} + 620067 p^{3} T^{4} - 303 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1035 T + 63873 T^{2} + 227096926 T^{3} + 63873 p^{3} T^{4} - 1035 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 510 T + 1098285 T^{2} + 353629860 T^{3} + 1098285 p^{3} T^{4} + 510 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 447 T + 657111 T^{2} - 269206770 T^{3} + 657111 p^{3} T^{4} - 447 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 777 T + 14268 p T^{2} - 780182781 T^{3} + 14268 p^{4} T^{4} - 777 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 78 T + 882717 T^{2} + 176745340 T^{3} + 882717 p^{3} T^{4} + 78 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 324 T + 1606299 T^{2} - 549274824 T^{3} + 1606299 p^{3} T^{4} - 324 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1191 T + 2555007 T^{2} - 2142250738 T^{3} + 2555007 p^{3} T^{4} - 1191 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.537774972035303840631030684562, −7.907583608186401647455822067093, −7.895331629371019626926421929848, −7.83950516558325037752876989357, −7.24376618121188601251448817185, −7.02805181536773130156284914985, −6.82170614364800102278932853488, −6.38288981780421370951668714170, −6.22154693509243648189103404191, −5.91508107824151688638658826860, −5.29806262730187595240166070137, −5.06387402262939474356702306310, −4.94382816920704298364905086955, −4.37609033524121752503905947276, −4.28668730939271111942249872291, −4.17913637251256716492393273133, −3.30577056430641318174548714212, −3.27158258157935411961709577307, −3.06606021409085581150303377446, −2.32853990848964681019277700029, −2.04844775726154793262508172964, −1.87588585337426722698519022384, −0.906607480859324615864999803966, −0.70781745139631214253111536534, −0.35892793162557326002364945228,
0.35892793162557326002364945228, 0.70781745139631214253111536534, 0.906607480859324615864999803966, 1.87588585337426722698519022384, 2.04844775726154793262508172964, 2.32853990848964681019277700029, 3.06606021409085581150303377446, 3.27158258157935411961709577307, 3.30577056430641318174548714212, 4.17913637251256716492393273133, 4.28668730939271111942249872291, 4.37609033524121752503905947276, 4.94382816920704298364905086955, 5.06387402262939474356702306310, 5.29806262730187595240166070137, 5.91508107824151688638658826860, 6.22154693509243648189103404191, 6.38288981780421370951668714170, 6.82170614364800102278932853488, 7.02805181536773130156284914985, 7.24376618121188601251448817185, 7.83950516558325037752876989357, 7.895331629371019626926421929848, 7.907583608186401647455822067093, 8.537774972035303840631030684562
