| L(s) = 1 | − 2·5-s − 4·11-s − 2·17-s − 4·19-s − 8·23-s − 25-s − 6·29-s + 8·31-s − 6·37-s − 6·41-s − 4·43-s − 7·49-s + 2·53-s + 8·55-s − 4·59-s − 2·61-s − 4·67-s − 8·71-s − 10·73-s + 8·79-s + 4·83-s + 4·85-s − 6·89-s + 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.20·11-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s − 49-s + 0.274·53-s + 1.07·55-s − 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s + 0.439·83-s + 0.433·85-s − 0.635·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.99757749866486, −15.32964808150714, −15.11246064320381, −14.40239851236580, −13.57334543565085, −13.39922089894708, −12.68868139298271, −12.00434434170607, −11.78464633445677, −11.03514914395727, −10.46360805572590, −10.10096349784598, −9.369781462314316, −8.525525288099009, −8.164389703758823, −7.745653438040241, −7.060134581080893, −6.369579148952002, −5.780054615909326, −5.006441000350352, −4.410848943757797, −3.832661908305993, −3.118838753052904, −2.307109076635718, −1.622834083754368, 0, 0,
1.622834083754368, 2.307109076635718, 3.118838753052904, 3.832661908305993, 4.410848943757797, 5.006441000350352, 5.780054615909326, 6.369579148952002, 7.060134581080893, 7.745653438040241, 8.164389703758823, 8.525525288099009, 9.369781462314316, 10.10096349784598, 10.46360805572590, 11.03514914395727, 11.78464633445677, 12.00434434170607, 12.68868139298271, 13.39922089894708, 13.57334543565085, 14.40239851236580, 15.11246064320381, 15.32964808150714, 15.99757749866486
