| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 2·13-s + 2·15-s − 2·17-s − 4·19-s + 21-s − 25-s − 27-s + 6·29-s + 2·35-s − 6·37-s − 2·39-s + 6·41-s − 8·43-s − 2·45-s − 8·47-s + 49-s + 2·51-s − 6·53-s + 4·57-s − 12·59-s + 10·61-s − 63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.338·35-s − 0.986·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 1.28·61-s − 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.53983730486236, −12.79979392712629, −12.56548455407855, −12.09379971265627, −11.57303186833168, −11.10648512504724, −10.80438054218897, −10.29701220849615, −9.635058741425648, −9.318881009819699, −8.494599851957504, −8.186399507817426, −7.861750155093877, −6.921205477457895, −6.695052172929108, −6.318788279581512, −5.627646347841864, −4.978777442677252, −4.600815284262503, −3.854132260846089, −3.650215083292858, −2.876446587216179, −2.153309616381419, −1.460105742553818, −0.6167841553433681, 0,
0.6167841553433681, 1.460105742553818, 2.153309616381419, 2.876446587216179, 3.650215083292858, 3.854132260846089, 4.600815284262503, 4.978777442677252, 5.627646347841864, 6.318788279581512, 6.695052172929108, 6.921205477457895, 7.861750155093877, 8.186399507817426, 8.494599851957504, 9.318881009819699, 9.635058741425648, 10.29701220849615, 10.80438054218897, 11.10648512504724, 11.57303186833168, 12.09379971265627, 12.56548455407855, 12.79979392712629, 13.53983730486236
