| L(s) = 1 | + 0.554·2-s + 3.04·3-s − 1.69·4-s + 1.69·6-s + 7-s − 2.04·8-s + 6.29·9-s − 2.44·11-s − 5.15·12-s − 5.96·13-s + 0.554·14-s + 2.24·16-s − 4.93·17-s + 3.49·18-s − 3.69·19-s + 3.04·21-s − 1.35·22-s + 23-s − 6.24·24-s − 3.30·26-s + 10.0·27-s − 1.69·28-s − 6.71·29-s − 2.80·31-s + 5.34·32-s − 7.45·33-s − 2.74·34-s + ⋯ |
| L(s) = 1 | + 0.392·2-s + 1.76·3-s − 0.846·4-s + 0.690·6-s + 0.377·7-s − 0.724·8-s + 2.09·9-s − 0.737·11-s − 1.48·12-s − 1.65·13-s + 0.148·14-s + 0.561·16-s − 1.19·17-s + 0.823·18-s − 0.847·19-s + 0.665·21-s − 0.289·22-s + 0.208·23-s − 1.27·24-s − 0.648·26-s + 1.93·27-s − 0.319·28-s − 1.24·29-s − 0.503·31-s + 0.944·32-s − 1.29·33-s − 0.470·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.554T + 2T^{2} \) |
| 3 | \( 1 - 3.04T + 3T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 5.96T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 + 1.33T + 37T^{2} \) |
| 41 | \( 1 - 6.87T + 41T^{2} \) |
| 43 | \( 1 + 0.554T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 6.13T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 0.948T + 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.102894225367344336857283854338, −7.60106236153447593219611580103, −6.88624394833029820502249280571, −5.62534245099284999875883621989, −4.68638764437823261678484854732, −4.33074518946051562218341403003, −3.36997547183104508126559043005, −2.55375512934309413234165045729, −1.92346582638956421991183983702, 0,
1.92346582638956421991183983702, 2.55375512934309413234165045729, 3.36997547183104508126559043005, 4.33074518946051562218341403003, 4.68638764437823261678484854732, 5.62534245099284999875883621989, 6.88624394833029820502249280571, 7.60106236153447593219611580103, 8.102894225367344336857283854338
