| L(s) = 1 | + 3·4-s − 2·5-s − 9-s − 12·11-s + 5·16-s − 12·19-s − 6·20-s − 25-s + 4·29-s + 20·31-s − 3·36-s − 4·41-s − 36·44-s + 2·45-s + 24·55-s − 16·59-s + 4·61-s + 3·64-s + 20·71-s − 36·76-s − 8·79-s − 10·80-s + 81-s + 12·89-s + 24·95-s + 12·99-s − 3·100-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s − 1/3·9-s − 3.61·11-s + 5/4·16-s − 2.75·19-s − 1.34·20-s − 1/5·25-s + 0.742·29-s + 3.59·31-s − 1/2·36-s − 0.624·41-s − 5.42·44-s + 0.298·45-s + 3.23·55-s − 2.08·59-s + 0.512·61-s + 3/8·64-s + 2.37·71-s − 4.12·76-s − 0.900·79-s − 1.11·80-s + 1/9·81-s + 1.27·89-s + 2.46·95-s + 1.20·99-s − 0.299·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013321091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013321091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.65432352920307798340698525594, −10.26264449642343933639969614141, −10.24954674872127833581824633607, −9.390474647614821434448921376422, −8.390750672063355757797823404983, −8.262007300807147454324838074952, −8.118093345649354850343782386613, −7.71382622968980540644197573214, −7.18107025616131508097129846225, −6.50068627462598728544158889301, −6.39823248084765288028849059082, −5.79385320903557292475169219793, −5.20980453827603346495185123115, −4.50967170295167665734040054949, −4.49572321492708365576392047484, −3.23140721448490455171248779770, −2.92714915453075174170310452548, −2.32101591337753869525714945934, −2.15426545704181043667133261772, −0.46777483556197831922160617880,
0.46777483556197831922160617880, 2.15426545704181043667133261772, 2.32101591337753869525714945934, 2.92714915453075174170310452548, 3.23140721448490455171248779770, 4.49572321492708365576392047484, 4.50967170295167665734040054949, 5.20980453827603346495185123115, 5.79385320903557292475169219793, 6.39823248084765288028849059082, 6.50068627462598728544158889301, 7.18107025616131508097129846225, 7.71382622968980540644197573214, 8.118093345649354850343782386613, 8.262007300807147454324838074952, 8.390750672063355757797823404983, 9.390474647614821434448921376422, 10.24954674872127833581824633607, 10.26264449642343933639969614141, 10.65432352920307798340698525594
