| L(s) = 1 | + 4·3-s + 2·5-s − 2·7-s + 10·9-s − 4·11-s + 4·13-s + 8·15-s + 2·17-s − 2·19-s − 8·21-s + 4·23-s − 6·25-s + 20·27-s + 12·29-s − 6·31-s − 16·33-s − 4·35-s − 2·37-s + 16·39-s + 6·41-s − 2·43-s + 20·45-s − 6·47-s + 4·49-s + 8·51-s − 4·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.894·5-s − 0.755·7-s + 10/3·9-s − 1.20·11-s + 1.10·13-s + 2.06·15-s + 0.485·17-s − 0.458·19-s − 1.74·21-s + 0.834·23-s − 6/5·25-s + 3.84·27-s + 2.22·29-s − 1.07·31-s − 2.78·33-s − 0.676·35-s − 0.328·37-s + 2.56·39-s + 0.937·41-s − 0.304·43-s + 2.98·45-s − 0.875·47-s + 4/7·49-s + 1.12·51-s − 0.549·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.75488471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.75488471\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 2 T + 2 p T^{2} - 8 T^{3} + 42 T^{4} - 8 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 2 T - 18 T^{3} - 26 T^{4} - 18 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 4 T^{2} + 84 T^{3} - 142 T^{4} + 84 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 2 T + 48 T^{2} + 54 T^{3} + 1174 T^{4} + 54 p T^{5} + 48 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 4 T + 28 T^{2} + 104 T^{3} - 282 T^{4} + 104 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 12 T + 140 T^{2} - 990 T^{3} + 6330 T^{4} - 990 p T^{5} + 140 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 6 T + 126 T^{2} + 548 T^{3} + 5886 T^{4} + 548 p T^{5} + 126 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 2 T + 60 T^{2} + 110 T^{3} + 1670 T^{4} + 110 p T^{5} + 60 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 6 T + 28 T^{2} - 254 T^{3} + 2806 T^{4} - 254 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 2 T + 68 T^{2} - 396 T^{3} + 1102 T^{4} - 396 p T^{5} + 68 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 6 T - 42 T^{3} + 350 T^{4} - 42 p T^{5} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 4 T + 172 T^{2} + 12 p T^{3} + 12662 T^{4} + 12 p^{2} T^{5} + 172 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 4 T + 196 T^{2} - 532 T^{3} + 16038 T^{4} - 532 p T^{5} + 196 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 22 T + 380 T^{2} - 4122 T^{3} + 38086 T^{4} - 4122 p T^{5} + 380 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 6 T + 114 T^{2} + 724 T^{3} + 10494 T^{4} + 724 p T^{5} + 114 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 14 T + 264 T^{2} + 2142 T^{3} + 25102 T^{4} + 2142 p T^{5} + 264 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6 T + 252 T^{2} - 1174 T^{3} + 26670 T^{4} - 1174 p T^{5} + 252 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 14 T + 216 T^{2} + 1628 T^{3} + 18254 T^{4} + 1628 p T^{5} + 216 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 100 T^{2} + 944 T^{3} + 2566 T^{4} + 944 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 44 T + 1026 T^{2} - 15702 T^{3} + 173146 T^{4} - 15702 p T^{5} + 1026 p^{2} T^{6} - 44 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 18 T + 156 T^{2} + 986 T^{3} - 19626 T^{4} + 986 p T^{5} + 156 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.56969436841309469676741268991, −5.27562464610079898638171518491, −5.25160685033564417152365619378, −5.12935160648749481535129426848, −4.80733807640413162702697218669, −4.51976399116400846493843443328, −4.34217731999479874058397999208, −4.26816339361242819182176339197, −4.14905811427844571085848291157, −3.55704810542718521253414570612, −3.46886979258412074696017677891, −3.45509371234327298303136088529, −3.44273569863235080748527270109, −3.01165092238520233557687253360, −2.78732804178158108722313458705, −2.78363994062778154911093608281, −2.32607864528845087923301960922, −2.16036146637275080956616887032, −2.01046334839465912463080077374, −1.97460818253951965966818265759, −1.60528101991303408216106110959, −1.18487163429069568135879779630, −0.977058343489906396051428939199, −0.72239371315323667810878725425, −0.34307159467423699625774946546,
0.34307159467423699625774946546, 0.72239371315323667810878725425, 0.977058343489906396051428939199, 1.18487163429069568135879779630, 1.60528101991303408216106110959, 1.97460818253951965966818265759, 2.01046334839465912463080077374, 2.16036146637275080956616887032, 2.32607864528845087923301960922, 2.78363994062778154911093608281, 2.78732804178158108722313458705, 3.01165092238520233557687253360, 3.44273569863235080748527270109, 3.45509371234327298303136088529, 3.46886979258412074696017677891, 3.55704810542718521253414570612, 4.14905811427844571085848291157, 4.26816339361242819182176339197, 4.34217731999479874058397999208, 4.51976399116400846493843443328, 4.80733807640413162702697218669, 5.12935160648749481535129426848, 5.25160685033564417152365619378, 5.27562464610079898638171518491, 5.56969436841309469676741268991
