| L(s) = 1 | + 14·4-s − 18·9-s − 184·11-s + 99·16-s + 216·19-s + 472·29-s − 120·31-s − 252·36-s + 88·41-s − 2.57e3·44-s − 98·49-s + 928·59-s − 1.36e3·61-s + 924·64-s − 1.48e3·71-s + 3.02e3·76-s + 816·79-s + 243·81-s + 2.66e3·89-s + 3.31e3·99-s + 136·101-s − 5.27e3·109-s + 6.60e3·116-s + 1.58e4·121-s − 1.68e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/4·4-s − 2/3·9-s − 5.04·11-s + 1.54·16-s + 2.60·19-s + 3.02·29-s − 0.695·31-s − 7/6·36-s + 0.335·41-s − 8.82·44-s − 2/7·49-s + 2.04·59-s − 2.87·61-s + 1.80·64-s − 2.47·71-s + 4.56·76-s + 1.16·79-s + 1/3·81-s + 3.17·89-s + 3.36·99-s + 0.133·101-s − 4.63·109-s + 5.28·116-s + 11.9·121-s − 1.21·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.142103983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.142103983\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 7 p T^{2} + 97 T^{4} - 7 p^{7} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 92 T + 4758 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5684 T^{2} + 17268502 T^{4} + 5684 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 676 T^{2} + 29648902 T^{4} + 676 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 108 T + 13254 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 284 p T^{2} + 101938534 T^{4} + 284 p^{7} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 236 T + 61982 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 60 T + 8462 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 176044 T^{2} + 12759471222 T^{4} - 176044 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 44 T + 106326 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 144940 T^{2} + 16379434678 T^{4} - 144940 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 53052 T^{2} - 317140666 T^{4} - 53052 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 521420 T^{2} + 112288959478 T^{4} - 521420 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 464 T + 458102 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 684 T + 535646 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 663244 T^{2} + 218042637142 T^{4} - 663244 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 740 T + 757502 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1317340 T^{2} + 723135691558 T^{4} - 1317340 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 408 T - 143586 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2031756 T^{2} + 1672847111702 T^{4} - 2031756 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1332 T + 1790774 T^{2} - 1332 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 580380 T^{2} + 1528544052038 T^{4} - 580380 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50474014685843443155313603229, −7.38996629216998087385113079209, −6.98237075153736080608079536876, −6.59731488510146135835771490082, −6.53484511589033305771407784021, −6.23086564069295230248777149662, −5.70485548349316412714677842163, −5.68003048887681946222963262919, −5.57058158395315353343262965099, −5.15256557484956982259026126728, −4.90853104921410795360070234082, −4.83976750361355594094720533691, −4.75149722176922339173200229711, −3.88900486469867854799395718820, −3.68159579613682151539571364965, −3.13253116104481272972588001508, −2.95761621223385487129699275954, −2.83528033427096463647666197159, −2.62745804493684440791573587046, −2.37999062387175313573331307629, −2.18498110468207537380224385567, −1.44632169346843767737681811884, −1.21100416868405288663222589763, −0.54152657936631692433730521784, −0.30721832406503930222481964623,
0.30721832406503930222481964623, 0.54152657936631692433730521784, 1.21100416868405288663222589763, 1.44632169346843767737681811884, 2.18498110468207537380224385567, 2.37999062387175313573331307629, 2.62745804493684440791573587046, 2.83528033427096463647666197159, 2.95761621223385487129699275954, 3.13253116104481272972588001508, 3.68159579613682151539571364965, 3.88900486469867854799395718820, 4.75149722176922339173200229711, 4.83976750361355594094720533691, 4.90853104921410795360070234082, 5.15256557484956982259026126728, 5.57058158395315353343262965099, 5.68003048887681946222963262919, 5.70485548349316412714677842163, 6.23086564069295230248777149662, 6.53484511589033305771407784021, 6.59731488510146135835771490082, 6.98237075153736080608079536876, 7.38996629216998087385113079209, 7.50474014685843443155313603229
