| L(s) = 1 | − 2·2-s + 4·4-s + 7.31·5-s + 17.3·7-s − 8·8-s − 14.6·10-s − 69.2·11-s − 3.25·13-s − 34.6·14-s + 16·16-s − 85.9·17-s − 19.9·19-s + 29.2·20-s + 138.·22-s − 23·23-s − 71.5·25-s + 6.50·26-s + 69.2·28-s + 47.1·29-s + 109.·31-s − 32·32-s + 171.·34-s + 126.·35-s + 86.9·37-s + 39.8·38-s − 58.5·40-s − 65.7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.654·5-s + 0.934·7-s − 0.353·8-s − 0.462·10-s − 1.89·11-s − 0.0694·13-s − 0.661·14-s + 0.250·16-s − 1.22·17-s − 0.240·19-s + 0.327·20-s + 1.34·22-s − 0.208·23-s − 0.572·25-s + 0.0491·26-s + 0.467·28-s + 0.301·29-s + 0.632·31-s − 0.176·32-s + 0.866·34-s + 0.611·35-s + 0.386·37-s + 0.170·38-s − 0.231·40-s − 0.250·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 7.31T + 125T^{2} \) |
| 7 | \( 1 - 17.3T + 343T^{2} \) |
| 11 | \( 1 + 69.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.25T + 2.19e3T^{2} \) |
| 17 | \( 1 + 85.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 47.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 109.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 86.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 65.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 164.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 631.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 665.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 83.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 969.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 462.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 382.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 817.T + 9.12e5T^{2} \) |
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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
−10.38736563868443191599140933450, −9.504800213098410368870686183925, −8.348867092084888479250315466615, −7.87726253472385184918461047146, −6.67091035029967032304593447353, −5.54454589810460713650879819794, −4.62503409436491508097606741070, −2.69751051312056910549723364037, −1.79049007682542678604872165044, 0,
1.79049007682542678604872165044, 2.69751051312056910549723364037, 4.62503409436491508097606741070, 5.54454589810460713650879819794, 6.67091035029967032304593447353, 7.87726253472385184918461047146, 8.348867092084888479250315466615, 9.504800213098410368870686183925, 10.38736563868443191599140933450
