| L(s) = 1 | + 0.814·2-s − 1.30·3-s − 31.3·4-s − 25·5-s − 1.06·6-s + 233.·7-s − 51.6·8-s − 241.·9-s − 20.3·10-s − 121·11-s + 40.9·12-s − 526.·13-s + 190.·14-s + 32.6·15-s + 960.·16-s + 824.·17-s − 196.·18-s − 361·19-s + 783.·20-s − 305.·21-s − 98.6·22-s + 1.29e3·23-s + 67.4·24-s + 625·25-s − 429.·26-s + 633.·27-s − 7.32e3·28-s + ⋯ |
| L(s) = 1 | + 0.144·2-s − 0.0838·3-s − 0.979·4-s − 0.447·5-s − 0.0120·6-s + 1.80·7-s − 0.285·8-s − 0.992·9-s − 0.0644·10-s − 0.301·11-s + 0.0821·12-s − 0.864·13-s + 0.259·14-s + 0.0375·15-s + 0.938·16-s + 0.691·17-s − 0.143·18-s − 0.229·19-s + 0.437·20-s − 0.151·21-s − 0.0434·22-s + 0.510·23-s + 0.0239·24-s + 0.200·25-s − 0.124·26-s + 0.167·27-s − 1.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 0.814T + 32T^{2} \) |
| 3 | \( 1 + 1.30T + 243T^{2} \) |
| 7 | \( 1 - 233.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 526.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 824.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.03e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.48e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.13e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.63e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.02e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.03e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.62e4T + 8.58e9T^{2} \) |
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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
−8.690199079101188209570390126457, −7.929883936803371680435939228635, −7.51942850271600496049421745809, −5.88048053368380564052057298886, −5.10253470065536989794554003145, −4.64518951905157019327130436084, −3.55872578992289377596943456046, −2.36574768123830089759131346189, −1.05507687857460935404821716020, 0,
1.05507687857460935404821716020, 2.36574768123830089759131346189, 3.55872578992289377596943456046, 4.64518951905157019327130436084, 5.10253470065536989794554003145, 5.88048053368380564052057298886, 7.51942850271600496049421745809, 7.929883936803371680435939228635, 8.690199079101188209570390126457
