| L(s) = 1 | + 4·5-s + 8·11-s − 4·19-s + 8·25-s + 8·29-s − 24·31-s − 8·41-s + 26·49-s + 32·55-s − 4·61-s + 56·71-s + 4·79-s + 48·89-s − 16·95-s − 48·101-s + 8·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s − 96·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2.41·11-s − 0.917·19-s + 8/5·25-s + 1.48·29-s − 4.31·31-s − 1.24·41-s + 26/7·49-s + 4.31·55-s − 0.512·61-s + 6.64·71-s + 0.450·79-s + 5.08·89-s − 1.64·95-s − 4.77·101-s + 8/11·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s − 7.71·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.252853700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.252853700\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T^{2} + 243 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 2258 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 56 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 963 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 14738 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 26 T^{2} - 453 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 28 T + 332 T^{2} - 28 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 242 T^{2} + 25203 T^{4} - 242 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 14978 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 99 p T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.63259356842618982602840815918, −6.19534137400881811775742925325, −6.05101935652253456404780789426, −5.88314432004906551456042125271, −5.57777414224293424254910665570, −5.40727598594747337360169853104, −5.26224297298396485058583604444, −5.07217323040412694251422009480, −4.91999556554107270355234591418, −4.49416212021678944730947641950, −4.18477254288311545857751797665, −3.97069088180782796571984107003, −3.93079644221829061736592331094, −3.59485102168089228962210265794, −3.52366781218182011922127880915, −3.18868937760313204037451650021, −2.82974143944792544314406388120, −2.39506595567585262386408950894, −2.20481051233391801090190591233, −2.00266310491874004032795401177, −1.86694560594946253072310613971, −1.61893625802541340779890286101, −1.12032246271454199030033261104, −0.814097246557516346549951117762, −0.52696369560528001664331915995,
0.52696369560528001664331915995, 0.814097246557516346549951117762, 1.12032246271454199030033261104, 1.61893625802541340779890286101, 1.86694560594946253072310613971, 2.00266310491874004032795401177, 2.20481051233391801090190591233, 2.39506595567585262386408950894, 2.82974143944792544314406388120, 3.18868937760313204037451650021, 3.52366781218182011922127880915, 3.59485102168089228962210265794, 3.93079644221829061736592331094, 3.97069088180782796571984107003, 4.18477254288311545857751797665, 4.49416212021678944730947641950, 4.91999556554107270355234591418, 5.07217323040412694251422009480, 5.26224297298396485058583604444, 5.40727598594747337360169853104, 5.57777414224293424254910665570, 5.88314432004906551456042125271, 6.05101935652253456404780789426, 6.19534137400881811775742925325, 6.63259356842618982602840815918
