| L(s) = 1 | + 2-s + 2.79·3-s + 4-s − 3.79·5-s + 2.79·6-s − 2·7-s + 8-s + 4.79·9-s − 3.79·10-s − 3.79·11-s + 2.79·12-s + 0.791·13-s − 2·14-s − 10.5·15-s + 16-s − 1.58·17-s + 4.79·18-s − 7.58·19-s − 3.79·20-s − 5.58·21-s − 3.79·22-s + 0.791·23-s + 2.79·24-s + 9.37·25-s + 0.791·26-s + 4.99·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.61·3-s + 0.5·4-s − 1.69·5-s + 1.13·6-s − 0.755·7-s + 0.353·8-s + 1.59·9-s − 1.19·10-s − 1.14·11-s + 0.805·12-s + 0.219·13-s − 0.534·14-s − 2.73·15-s + 0.250·16-s − 0.383·17-s + 1.12·18-s − 1.73·19-s − 0.847·20-s − 1.21·21-s − 0.808·22-s + 0.164·23-s + 0.569·24-s + 1.87·25-s + 0.155·26-s + 0.962·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 - 0.791T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 + 7.58T + 19T^{2} \) |
| 23 | \( 1 - 0.791T + 23T^{2} \) |
| 29 | \( 1 - 0.791T + 29T^{2} \) |
| 31 | \( 1 + 5.37T + 31T^{2} \) |
| 41 | \( 1 + 5.20T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 - 7.58T + 53T^{2} \) |
| 59 | \( 1 + 7.58T + 59T^{2} \) |
| 61 | \( 1 + 8.20T + 61T^{2} \) |
| 67 | \( 1 - 7.37T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 4.41T + 97T^{2} \) |
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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
−8.430049700660278996550799263120, −7.72171211054302389873152461519, −7.14933880785220646855591311992, −6.34643379037890045416204545226, −4.95618354491442827153290424462, −4.16075339788162996450092463231, −3.55079034813787560044898625385, −2.96288475815850050195052766523, −2.07383749064450879319630605217, 0,
2.07383749064450879319630605217, 2.96288475815850050195052766523, 3.55079034813787560044898625385, 4.16075339788162996450092463231, 4.95618354491442827153290424462, 6.34643379037890045416204545226, 7.14933880785220646855591311992, 7.72171211054302389873152461519, 8.430049700660278996550799263120
