| L(s) = 1 | + 1.54·2-s + 0.697·3-s + 0.401·4-s + 0.276·5-s + 1.08·6-s + 3.52·7-s − 2.47·8-s − 2.51·9-s + 0.428·10-s + 11-s + 0.279·12-s − 2.68·13-s + 5.46·14-s + 0.192·15-s − 4.64·16-s − 1.96·17-s − 3.89·18-s + 2.84·19-s + 0.110·20-s + 2.46·21-s + 1.54·22-s + 2.68·23-s − 1.72·24-s − 4.92·25-s − 4.15·26-s − 3.84·27-s + 1.41·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.402·3-s + 0.200·4-s + 0.123·5-s + 0.441·6-s + 1.33·7-s − 0.875·8-s − 0.837·9-s + 0.135·10-s + 0.301·11-s + 0.0808·12-s − 0.744·13-s + 1.46·14-s + 0.0497·15-s − 1.16·16-s − 0.476·17-s − 0.917·18-s + 0.652·19-s + 0.0247·20-s + 0.536·21-s + 0.330·22-s + 0.559·23-s − 0.352·24-s − 0.984·25-s − 0.815·26-s − 0.740·27-s + 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 - 0.697T + 3T^{2} \) |
| 5 | \( 1 - 0.276T + 5T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 17 | \( 1 + 1.96T + 17T^{2} \) |
| 19 | \( 1 - 2.84T + 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 - 5.73T + 41T^{2} \) |
| 43 | \( 1 - 4.46T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 + 7.77T + 53T^{2} \) |
| 59 | \( 1 + 2.10T + 59T^{2} \) |
| 61 | \( 1 + 7.14T + 61T^{2} \) |
| 67 | \( 1 - 3.64T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 4.78T + 79T^{2} \) |
| 83 | \( 1 + 1.15T + 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 + 1.22T + 97T^{2} \) |
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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
−7.42487422386343339977255851129, −6.51666291992690290163837955881, −5.72427837055926828267012563771, −5.19336388151866625391474828411, −4.67097109493216423506256340974, −3.92341789399058803299236481273, −3.13937315541030973409777361894, −2.41569645631137037510990571912, −1.56642161517784452674033265796, 0,
1.56642161517784452674033265796, 2.41569645631137037510990571912, 3.13937315541030973409777361894, 3.92341789399058803299236481273, 4.67097109493216423506256340974, 5.19336388151866625391474828411, 5.72427837055926828267012563771, 6.51666291992690290163837955881, 7.42487422386343339977255851129
