| L(s) = 1 | − 2.59·2-s + 5.80·3-s − 1.24·4-s − 7.55·5-s − 15.0·6-s + 7·7-s + 24.0·8-s + 6.72·9-s + 19.6·10-s − 32.2·11-s − 7.24·12-s − 31.5·13-s − 18.1·14-s − 43.9·15-s − 52.4·16-s + 20.7·17-s − 17.4·18-s − 64.8·19-s + 9.43·20-s + 40.6·21-s + 83.8·22-s − 23·23-s + 139.·24-s − 67.8·25-s + 81.9·26-s − 117.·27-s − 8.73·28-s + ⋯ |
| L(s) = 1 | − 0.918·2-s + 1.11·3-s − 0.155·4-s − 0.676·5-s − 1.02·6-s + 0.377·7-s + 1.06·8-s + 0.249·9-s + 0.621·10-s − 0.884·11-s − 0.174·12-s − 0.673·13-s − 0.347·14-s − 0.755·15-s − 0.819·16-s + 0.296·17-s − 0.228·18-s − 0.783·19-s + 0.105·20-s + 0.422·21-s + 0.812·22-s − 0.208·23-s + 1.18·24-s − 0.542·25-s + 0.618·26-s − 0.839·27-s − 0.0589·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 2.59T + 8T^{2} \) |
| 3 | \( 1 - 5.80T + 27T^{2} \) |
| 5 | \( 1 + 7.55T + 125T^{2} \) |
| 11 | \( 1 + 32.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 33.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 46.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 383.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 120.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 640.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 616.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 483.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 862.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 313.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 523.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 930.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 12.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 255.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 306.T + 9.12e5T^{2} \) |
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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
−11.80122367853324877737654252846, −10.55887912895033609390463633760, −9.708350427645078208614130299181, −8.549028511037897231809960484370, −8.113854480827379320450720249486, −7.24724736914142766901994889285, −5.07862093478304879857072719133, −3.72589997974349443797012457721, −2.12747082233323676914934424053, 0,
2.12747082233323676914934424053, 3.72589997974349443797012457721, 5.07862093478304879857072719133, 7.24724736914142766901994889285, 8.113854480827379320450720249486, 8.549028511037897231809960484370, 9.708350427645078208614130299181, 10.55887912895033609390463633760, 11.80122367853324877737654252846
