| L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s + 1.61·6-s + 2·7-s − 2.23·8-s + 9-s − 0.763·11-s + 0.618·12-s + 13-s + 3.23·14-s − 4.85·16-s − 5.47·17-s + 1.61·18-s − 6.23·19-s + 2·21-s − 1.23·22-s + 6.47·23-s − 2.23·24-s + 1.61·26-s + 27-s + 1.23·28-s − 3.23·29-s − 6·31-s − 3.38·32-s − 0.763·33-s − 8.85·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.660·6-s + 0.755·7-s − 0.790·8-s + 0.333·9-s − 0.230·11-s + 0.178·12-s + 0.277·13-s + 0.864·14-s − 1.21·16-s − 1.32·17-s + 0.381·18-s − 1.43·19-s + 0.436·21-s − 0.263·22-s + 1.34·23-s − 0.456·24-s + 0.317·26-s + 0.192·27-s + 0.233·28-s − 0.600·29-s − 1.07·31-s − 0.597·32-s − 0.132·33-s − 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 + 0.763T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 2.76T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−7.34954707772589817417225807981, −6.67941045769765296863226756595, −6.03400845217464075315126936096, −5.07150435215031521292060590305, −4.70059254549793649310868406102, −3.97331145839338364719124844134, −3.28531385265656647994357362455, −2.39501068571879020484314113277, −1.68640141141037256865073922142, 0,
1.68640141141037256865073922142, 2.39501068571879020484314113277, 3.28531385265656647994357362455, 3.97331145839338364719124844134, 4.70059254549793649310868406102, 5.07150435215031521292060590305, 6.03400845217464075315126936096, 6.67941045769765296863226756595, 7.34954707772589817417225807981
