| L(s) = 1 | + 2-s − 1.61·3-s + 4-s − 1.38·5-s − 1.61·6-s + 0.236·7-s + 8-s − 0.381·9-s − 1.38·10-s − 4.23·11-s − 1.61·12-s − 13-s + 0.236·14-s + 2.23·15-s + 16-s − 4.23·17-s − 0.381·18-s + 2·19-s − 1.38·20-s − 0.381·21-s − 4.23·22-s − 3.76·23-s − 1.61·24-s − 3.09·25-s − 26-s + 5.47·27-s + 0.236·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.934·3-s + 0.5·4-s − 0.618·5-s − 0.660·6-s + 0.0892·7-s + 0.353·8-s − 0.127·9-s − 0.437·10-s − 1.27·11-s − 0.467·12-s − 0.277·13-s + 0.0630·14-s + 0.577·15-s + 0.250·16-s − 1.02·17-s − 0.0900·18-s + 0.458·19-s − 0.309·20-s − 0.0833·21-s − 0.903·22-s − 0.784·23-s − 0.330·24-s − 0.618·25-s − 0.196·26-s + 1.05·27-s + 0.0446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 + 1.47T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 + 0.145T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 0.763T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−10.75550022492265920355365148009, −9.779310370633365036992065654641, −8.340542059575305887345664585683, −7.55313253354201825755994795926, −6.52286239153267665937487243535, −5.53522707827406947590541907383, −4.88465287099634814764121847312, −3.72426930725569106217054384489, −2.34630354444356126931490073198, 0,
2.34630354444356126931490073198, 3.72426930725569106217054384489, 4.88465287099634814764121847312, 5.53522707827406947590541907383, 6.52286239153267665937487243535, 7.55313253354201825755994795926, 8.340542059575305887345664585683, 9.779310370633365036992065654641, 10.75550022492265920355365148009
