| L(s) = 1 | − 3-s − 2·4-s − 2·5-s − 4·7-s + 9-s + 5·11-s + 2·12-s − 4·13-s + 2·15-s + 4·16-s − 5·17-s − 2·19-s + 4·20-s + 4·21-s + 4·23-s − 25-s − 27-s + 8·28-s + 29-s − 5·31-s − 5·33-s + 8·35-s − 2·36-s − 7·37-s + 4·39-s − 41-s + 7·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.50·11-s + 0.577·12-s − 1.10·13-s + 0.516·15-s + 16-s − 1.21·17-s − 0.458·19-s + 0.894·20-s + 0.872·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.51·28-s + 0.185·29-s − 0.898·31-s − 0.870·33-s + 1.35·35-s − 1/3·36-s − 1.15·37-s + 0.640·39-s − 0.156·41-s + 1.06·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.65726821511143684863268263500, −12.19291964120732634230943912849, −10.87217891148406657550283695403, −9.547800136867235764059429479625, −8.968031421605497501048459827854, −7.27717120825502804586773539262, −6.27260516956401859469001094478, −4.61406346342867762230514888841, −3.61135250366573735073990623116, 0,
3.61135250366573735073990623116, 4.61406346342867762230514888841, 6.27260516956401859469001094478, 7.27717120825502804586773539262, 8.968031421605497501048459827854, 9.547800136867235764059429479625, 10.87217891148406657550283695403, 12.19291964120732634230943912849, 12.65726821511143684863268263500
