| L(s) = 1 | + 1.31·2-s + 2.16·3-s − 0.269·4-s + 2.84·6-s − 7-s − 2.98·8-s + 1.66·9-s + 2.63·11-s − 0.581·12-s − 6.78·13-s − 1.31·14-s − 3.38·16-s + 3.67·17-s + 2.19·18-s − 3.70·19-s − 2.16·21-s + 3.47·22-s + 23-s − 6.45·24-s − 8.93·26-s − 2.87·27-s + 0.269·28-s − 2.75·29-s − 8.90·31-s + 1.51·32-s + 5.70·33-s + 4.82·34-s + ⋯ |
| L(s) = 1 | + 0.930·2-s + 1.24·3-s − 0.134·4-s + 1.16·6-s − 0.377·7-s − 1.05·8-s + 0.556·9-s + 0.795·11-s − 0.167·12-s − 1.88·13-s − 0.351·14-s − 0.847·16-s + 0.890·17-s + 0.517·18-s − 0.850·19-s − 0.471·21-s + 0.740·22-s + 0.208·23-s − 1.31·24-s − 1.75·26-s − 0.553·27-s + 0.0508·28-s − 0.511·29-s − 1.59·31-s + 0.267·32-s + 0.992·33-s + 0.828·34-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 | \( 1 - 1.31T + 2T^{2} \) |
| 3 | \( 1 - 2.16T + 3T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 + 6.78T + 13T^{2} \) |
| 17 | \( 1 - 3.67T + 17T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 31 | \( 1 + 8.90T + 31T^{2} \) |
| 37 | \( 1 + 8.36T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 + 0.435T + 53T^{2} \) |
| 59 | \( 1 - 1.16T + 59T^{2} \) |
| 61 | \( 1 + 3.72T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 0.347T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 4.15T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 - 9.99T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
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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
−8.096554282197433104795124797328, −7.32912823464841404032539828308, −6.65534706491206093382773723336, −5.61574120359102978128698124993, −5.00365363736517282140711199085, −4.02420467358747769633057884269, −3.51248950805419906699399186439, −2.74614993195288608539066136816, −1.91695142253314689799608851221, 0,
1.91695142253314689799608851221, 2.74614993195288608539066136816, 3.51248950805419906699399186439, 4.02420467358747769633057884269, 5.00365363736517282140711199085, 5.61574120359102978128698124993, 6.65534706491206093382773723336, 7.32912823464841404032539828308, 8.096554282197433104795124797328
