| L(s) = 1 | − 2.62·2-s − 1.17·3-s + 4.90·4-s − 5-s + 3.08·6-s + 3.02·7-s − 7.62·8-s − 1.62·9-s + 2.62·10-s − 11-s − 5.75·12-s + 2.13·13-s − 7.95·14-s + 1.17·15-s + 10.2·16-s − 3.09·17-s + 4.26·18-s − 1.70·19-s − 4.90·20-s − 3.55·21-s + 2.62·22-s − 4.01·23-s + 8.94·24-s + 25-s − 5.61·26-s + 5.42·27-s + 14.8·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.677·3-s + 2.45·4-s − 0.447·5-s + 1.25·6-s + 1.14·7-s − 2.69·8-s − 0.541·9-s + 0.830·10-s − 0.301·11-s − 1.66·12-s + 0.593·13-s − 2.12·14-s + 0.302·15-s + 2.55·16-s − 0.750·17-s + 1.00·18-s − 0.392·19-s − 1.09·20-s − 0.774·21-s + 0.560·22-s − 0.836·23-s + 1.82·24-s + 0.200·25-s − 1.10·26-s + 1.04·27-s + 2.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 1.17T + 3T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 13 | \( 1 - 2.13T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 + 4.01T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 + 0.398T + 41T^{2} \) |
| 43 | \( 1 + 5.72T + 43T^{2} \) |
| 47 | \( 1 + 4.40T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 - 6.08T + 61T^{2} \) |
| 67 | \( 1 + 8.55T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 9.67T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 0.620T + 97T^{2} \) |
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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
−8.246202076965731368830801333807, −7.77706423408444667165519146844, −6.59036204312517447775099870657, −6.35958113645697795184452990633, −5.24394201172537467800723544174, −4.38085197931769292854339533985, −2.98090059779812544075049324294, −2.05041094885254768007076990800, −1.05686693806274252592964638764, 0,
1.05686693806274252592964638764, 2.05041094885254768007076990800, 2.98090059779812544075049324294, 4.38085197931769292854339533985, 5.24394201172537467800723544174, 6.35958113645697795184452990633, 6.59036204312517447775099870657, 7.77706423408444667165519146844, 8.246202076965731368830801333807
