| L(s) = 1 | − 2.05·2-s − 0.874·3-s + 2.23·4-s + 2.05·5-s + 1.79·6-s + 4.62·7-s − 0.480·8-s − 2.23·9-s − 4.23·10-s − 5.06·11-s − 1.95·12-s + 13-s − 9.51·14-s − 1.79·15-s − 3.47·16-s − 3.45·17-s + 4.59·18-s + 1.24·19-s + 4.59·20-s − 4.04·21-s + 10.4·22-s − 9.22·23-s + 0.419·24-s − 0.773·25-s − 2.05·26-s + 4.57·27-s + 10.3·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.504·3-s + 1.11·4-s + 0.919·5-s + 0.734·6-s + 1.74·7-s − 0.169·8-s − 0.745·9-s − 1.33·10-s − 1.52·11-s − 0.563·12-s + 0.277·13-s − 2.54·14-s − 0.464·15-s − 0.869·16-s − 0.838·17-s + 1.08·18-s + 0.284·19-s + 1.02·20-s − 0.882·21-s + 2.22·22-s − 1.92·23-s + 0.0857·24-s − 0.154·25-s − 0.403·26-s + 0.880·27-s + 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 3 | \( 1 + 0.874T + 3T^{2} \) |
| 5 | \( 1 - 2.05T + 5T^{2} \) |
| 7 | \( 1 - 4.62T + 7T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 9.22T + 23T^{2} \) |
| 29 | \( 1 + 7.54T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 + 0.239T + 37T^{2} \) |
| 41 | \( 1 - 5.84T + 41T^{2} \) |
| 43 | \( 1 - 2.02T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 7.33T + 53T^{2} \) |
| 59 | \( 1 + 8.29T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 + 6.03T + 67T^{2} \) |
| 71 | \( 1 + 2.52T + 71T^{2} \) |
| 73 | \( 1 - 3.09T + 73T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 + 1.83T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.417097606888023516577181737307, −8.760642856730709198421651648886, −7.83455921331813092866950588896, −7.58730975200969541571677414465, −5.99059784575628232311357575716, −5.47849517251692026267152161952, −4.45457666104382658444080264677, −2.34039543211002361660192642088, −1.71591269728763589972714583033, 0,
1.71591269728763589972714583033, 2.34039543211002361660192642088, 4.45457666104382658444080264677, 5.47849517251692026267152161952, 5.99059784575628232311357575716, 7.58730975200969541571677414465, 7.83455921331813092866950588896, 8.760642856730709198421651648886, 9.417097606888023516577181737307
