| L(s) = 1 | − 0.906·2-s − 1.17·4-s − 2.59·7-s + 2.88·8-s − 0.741·11-s + 3.78·13-s + 2.35·14-s − 0.258·16-s − 3.16·17-s − 19-s + 0.672·22-s + 0.570·23-s − 3.43·26-s + 3.05·28-s − 6·29-s + 5.83·31-s − 5.52·32-s + 2.87·34-s + 1.40·37-s + 0.906·38-s + 3.83·41-s + 2.59·43-s + 0.872·44-s − 0.517·46-s + 5.08·47-s − 0.258·49-s − 4.46·52-s + ⋯ |
| L(s) = 1 | − 0.641·2-s − 0.588·4-s − 0.981·7-s + 1.01·8-s − 0.223·11-s + 1.05·13-s + 0.629·14-s − 0.0647·16-s − 0.768·17-s − 0.229·19-s + 0.143·22-s + 0.119·23-s − 0.673·26-s + 0.577·28-s − 1.11·29-s + 1.04·31-s − 0.977·32-s + 0.492·34-s + 0.230·37-s + 0.147·38-s + 0.599·41-s + 0.395·43-s + 0.131·44-s − 0.0763·46-s + 0.741·47-s − 0.0369·49-s − 0.618·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.906T + 2T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 + 0.741T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 23 | \( 1 - 0.570T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 - 5.08T + 47T^{2} \) |
| 53 | \( 1 + 0.160T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 + 4.19T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 3.78T + 97T^{2} \) |
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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
−8.137199129394524723862955904665, −7.47429643642054681629524521769, −6.54517584741660624477279027144, −5.98782842413525368513123615886, −5.00001661066083811459830228697, −4.13579626074554309800000893590, −3.48132549283550199735641701122, −2.35837661969547337073244569692, −1.11349401184104800243792048397, 0,
1.11349401184104800243792048397, 2.35837661969547337073244569692, 3.48132549283550199735641701122, 4.13579626074554309800000893590, 5.00001661066083811459830228697, 5.98782842413525368513123615886, 6.54517584741660624477279027144, 7.47429643642054681629524521769, 8.137199129394524723862955904665
