| L(s) = 1 | − 7.10·2-s − 0.0987·3-s + 18.5·4-s + 0.702·6-s − 118.·7-s + 95.7·8-s − 242.·9-s − 426.·11-s − 1.83·12-s − 825.·13-s + 839.·14-s − 1.27e3·16-s − 1.73e3·17-s + 1.72e3·18-s + 1.59e3·19-s + 11.6·21-s + 3.03e3·22-s + 2.78e3·23-s − 9.45·24-s + 5.86e3·26-s + 47.9·27-s − 2.18e3·28-s + 1.97e3·29-s − 3.99e3·31-s + 5.98e3·32-s + 42.1·33-s + 1.23e4·34-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.00633·3-s + 0.579·4-s + 0.00796·6-s − 0.910·7-s + 0.528·8-s − 0.999·9-s − 1.06·11-s − 0.00366·12-s − 1.35·13-s + 1.14·14-s − 1.24·16-s − 1.45·17-s + 1.25·18-s + 1.01·19-s + 0.00576·21-s + 1.33·22-s + 1.09·23-s − 0.00335·24-s + 1.70·26-s + 0.0126·27-s − 0.527·28-s + 0.435·29-s − 0.747·31-s + 1.03·32-s + 0.00673·33-s + 1.83·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 7.10T + 32T^{2} \) |
| 3 | \( 1 + 0.0987T + 243T^{2} \) |
| 7 | \( 1 + 118.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 426.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 825.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.59e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.61e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.05e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 5.42e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.71e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 438.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.66e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.04e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.73e5T + 8.58e9T^{2} \) |
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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
−9.046056436112176565433884294381, −7.915091742768266673909226439731, −7.39847734282664181216299284426, −6.46948332343921715959063295668, −5.36135011209886505904015572866, −4.48708827066991702866611031915, −2.88680087676133797374446693917, −2.37592600833808456935761445708, −0.71278297242195994231088144655, 0,
0.71278297242195994231088144655, 2.37592600833808456935761445708, 2.88680087676133797374446693917, 4.48708827066991702866611031915, 5.36135011209886505904015572866, 6.46948332343921715959063295668, 7.39847734282664181216299284426, 7.915091742768266673909226439731, 9.046056436112176565433884294381
