| L(s) = 1 | − 0.305·2-s + 3-s − 1.90·4-s + 2.75·5-s − 0.305·6-s − 0.574·7-s + 1.19·8-s + 9-s − 0.842·10-s − 0.481·11-s − 1.90·12-s + 4.48·13-s + 0.175·14-s + 2.75·15-s + 3.44·16-s + 17-s − 0.305·18-s − 5.50·19-s − 5.25·20-s − 0.574·21-s + 0.147·22-s − 5.33·23-s + 1.19·24-s + 2.59·25-s − 1.37·26-s + 27-s + 1.09·28-s + ⋯ |
| L(s) = 1 | − 0.216·2-s + 0.577·3-s − 0.953·4-s + 1.23·5-s − 0.124·6-s − 0.217·7-s + 0.422·8-s + 0.333·9-s − 0.266·10-s − 0.145·11-s − 0.550·12-s + 1.24·13-s + 0.0469·14-s + 0.711·15-s + 0.861·16-s + 0.242·17-s − 0.0720·18-s − 1.26·19-s − 1.17·20-s − 0.125·21-s + 0.0313·22-s − 1.11·23-s + 0.243·24-s + 0.519·25-s − 0.268·26-s + 0.192·27-s + 0.207·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 0.305T + 2T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 7 | \( 1 + 0.574T + 7T^{2} \) |
| 11 | \( 1 + 0.481T + 11T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 19 | \( 1 + 5.50T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 + 8.37T + 29T^{2} \) |
| 31 | \( 1 - 4.27T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 + 7.93T + 47T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 2.08T + 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 - 0.129T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 - 6.88T + 89T^{2} \) |
| 97 | \( 1 - 0.861T + 97T^{2} \) |
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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
−7.85908001639355352067893070447, −6.61103719142753005583396727508, −6.13593388855289115148818023303, −5.45775849810622694699766060801, −4.63003278169310082169898583713, −3.83216669408817354264702795938, −3.20519391784001891795175878497, −1.97891327833891517377060603840, −1.49294022710755353217949725261, 0,
1.49294022710755353217949725261, 1.97891327833891517377060603840, 3.20519391784001891795175878497, 3.83216669408817354264702795938, 4.63003278169310082169898583713, 5.45775849810622694699766060801, 6.13593388855289115148818023303, 6.61103719142753005583396727508, 7.85908001639355352067893070447
