L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 5·11-s − 13-s + 15-s − 2·17-s − 6·19-s + 21-s + 9·23-s + 25-s + 5·27-s + 6·29-s − 5·31-s − 5·33-s + 35-s + 37-s + 39-s + 3·41-s + 6·43-s + 2·45-s − 7·47-s + 49-s + 2·51-s − 12·53-s − 5·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.485·17-s − 1.37·19-s + 0.218·21-s + 1.87·23-s + 1/5·25-s + 0.962·27-s + 1.11·29-s − 0.898·31-s − 0.870·33-s + 0.169·35-s + 0.164·37-s + 0.160·39-s + 0.468·41-s + 0.914·43-s + 0.298·45-s − 1.02·47-s + 1/7·49-s + 0.280·51-s − 1.64·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.278283368893772609223079196613, −7.27211159080710657510784057376, −6.44361966934794791680955647047, −6.26750413669250201515588865349, −5.02961690112152856334551402167, −4.42637788908097299066150393977, −3.49249730310852852773224541883, −2.61652812405999217097848225758, −1.23503925829553907016295769157, 0,
1.23503925829553907016295769157, 2.61652812405999217097848225758, 3.49249730310852852773224541883, 4.42637788908097299066150393977, 5.02961690112152856334551402167, 6.26750413669250201515588865349, 6.44361966934794791680955647047, 7.27211159080710657510784057376, 8.278283368893772609223079196613