| L(s) = 1 | − 0.756·2-s − 3.07·3-s − 1.42·4-s + 0.386·5-s + 2.32·6-s + 0.194·7-s + 2.59·8-s + 6.43·9-s − 0.292·10-s + 2.36·11-s + 4.38·12-s − 1.22·13-s − 0.147·14-s − 1.18·15-s + 0.894·16-s − 5.04·17-s − 4.87·18-s + 4.24·19-s − 0.551·20-s − 0.598·21-s − 1.78·22-s + 1.53·23-s − 7.96·24-s − 4.85·25-s + 0.927·26-s − 10.5·27-s − 0.278·28-s + ⋯ |
| L(s) = 1 | − 0.534·2-s − 1.77·3-s − 0.713·4-s + 0.172·5-s + 0.948·6-s + 0.0736·7-s + 0.916·8-s + 2.14·9-s − 0.0923·10-s + 0.712·11-s + 1.26·12-s − 0.340·13-s − 0.0394·14-s − 0.306·15-s + 0.223·16-s − 1.22·17-s − 1.14·18-s + 0.973·19-s − 0.123·20-s − 0.130·21-s − 0.381·22-s + 0.319·23-s − 1.62·24-s − 0.970·25-s + 0.181·26-s − 2.03·27-s − 0.0525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 503 | \( 1 + T \) |
| good | 2 | \( 1 + 0.756T + 2T^{2} \) |
| 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 0.386T + 5T^{2} \) |
| 7 | \( 1 - 0.194T + 7T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 - 3.33T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 + 1.08T + 47T^{2} \) |
| 53 | \( 1 - 1.55T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 + 5.45T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.902T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−10.44785986863865743805175267068, −9.728571855113006704074525538469, −8.958250027829183239630253920404, −7.60564557589694798257501436264, −6.74826869901705286779948571615, −5.72375640923234197221972978645, −4.90319260196671514686411914704, −4.02222312527322453501192208442, −1.47409392221607039132248049202, 0,
1.47409392221607039132248049202, 4.02222312527322453501192208442, 4.90319260196671514686411914704, 5.72375640923234197221972978645, 6.74826869901705286779948571615, 7.60564557589694798257501436264, 8.958250027829183239630253920404, 9.728571855113006704074525538469, 10.44785986863865743805175267068
