| L(s) = 1 | + 64·2-s + 3.07e3·4-s + 1.06e3·5-s − 2.33e4·7-s + 1.31e5·8-s + 6.83e4·10-s + 4.21e4·11-s + 1.79e6·13-s − 1.49e6·14-s + 5.24e6·16-s + 7.34e6·17-s + 4.63e6·19-s + 3.28e6·20-s + 2.69e6·22-s + 3.82e7·23-s − 6.51e7·25-s + 1.15e8·26-s − 7.15e7·28-s + 9.46e7·29-s + 1.15e8·31-s + 2.01e8·32-s + 4.70e8·34-s − 2.48e7·35-s + 5.79e8·37-s + 2.96e8·38-s + 1.39e8·40-s − 7.45e8·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.152·5-s − 0.523·7-s + 1.41·8-s + 0.216·10-s + 0.0789·11-s + 1.34·13-s − 0.741·14-s + 5/4·16-s + 1.25·17-s + 0.429·19-s + 0.229·20-s + 0.111·22-s + 1.24·23-s − 1.33·25-s + 1.90·26-s − 0.785·28-s + 0.857·29-s + 0.725·31-s + 1.06·32-s + 1.77·34-s − 0.0800·35-s + 1.37·37-s + 0.607·38-s + 0.216·40-s − 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(10.99744651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.99744651\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 1068 T + 13259249 p T^{2} - 1068 p^{11} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 23300 T + 37330215 p T^{2} + 23300 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 42150 T + 485498981503 T^{2} - 42150 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1799800 T + 3128334358074 T^{2} - 1799800 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7347456 T + 50394909033250 T^{2} - 7347456 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4639288 T + 234484733860674 T^{2} - 4639288 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1664436 p T + 1906179871616050 T^{2} - 1664436 p^{12} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 94666800 T + 8489726824469158 T^{2} - 94666800 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 115716364 T + 16988559225154161 T^{2} - 115716364 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 579266500 T + 275090346358613502 T^{2} - 579266500 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 745710600 T + 1233612390052180606 T^{2} + 745710600 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2326180600 T + 2836617323237698290 T^{2} - 2326180600 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1036094172 T + 174749937120116002 T^{2} + 1036094172 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2552794308 T + 11696583253676167285 T^{2} + 2552794308 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6425729400 T + 69412435347846294742 T^{2} + 6425729400 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18991708192 T + \)\(17\!\cdots\!38\)\( T^{2} - 18991708192 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24200045800 T + \)\(38\!\cdots\!42\)\( T^{2} - 24200045800 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 833738400 T + \)\(37\!\cdots\!42\)\( T^{2} + 833738400 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9588012250 T + \)\(32\!\cdots\!23\)\( T^{2} - 9588012250 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 63000343000 T + \)\(24\!\cdots\!58\)\( T^{2} - 63000343000 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5110325670 T + \)\(18\!\cdots\!59\)\( T^{2} - 5110325670 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 79532613000 T + \)\(39\!\cdots\!02\)\( T^{2} + 79532613000 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 44619539950 T + \)\(54\!\cdots\!75\)\( T^{2} - 44619539950 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.14448951424210367466377471759, −12.98988309484962231961773676605, −12.08565836311412545876394946310, −11.88095785004825033221397819269, −10.89011701657088119588275266225, −10.83251518818926212315067536395, −9.590518790621484461226004547015, −9.553451890079147389388619493616, −8.096362662568501135370283139701, −7.988758556483337198051423398254, −6.67959319619644962673688686325, −6.60716140602176337564670109330, −5.56237545843303703708449782742, −5.40403721598051539048357295963, −4.25634418108138598079911165658, −3.75028544895859769946466084087, −3.06341225230493165006869343877, −2.44592813371377458741256208325, −1.29312096149298257788083348636, −0.825267027694447494269986852694,
0.825267027694447494269986852694, 1.29312096149298257788083348636, 2.44592813371377458741256208325, 3.06341225230493165006869343877, 3.75028544895859769946466084087, 4.25634418108138598079911165658, 5.40403721598051539048357295963, 5.56237545843303703708449782742, 6.60716140602176337564670109330, 6.67959319619644962673688686325, 7.988758556483337198051423398254, 8.096362662568501135370283139701, 9.553451890079147389388619493616, 9.590518790621484461226004547015, 10.83251518818926212315067536395, 10.89011701657088119588275266225, 11.88095785004825033221397819269, 12.08565836311412545876394946310, 12.98988309484962231961773676605, 13.14448951424210367466377471759
