L(s) = 1 | + 1.85·2-s − 3-s + 1.42·4-s + 1.52·5-s − 1.85·6-s + 0.592·7-s − 1.05·8-s + 9-s + 2.81·10-s + 5.80·11-s − 1.42·12-s + 0.790·13-s + 1.09·14-s − 1.52·15-s − 4.81·16-s − 17-s + 1.85·18-s − 8.36·19-s + 2.17·20-s − 0.592·21-s + 10.7·22-s − 4.55·23-s + 1.05·24-s − 2.68·25-s + 1.46·26-s − 27-s + 0.847·28-s + ⋯ |
L(s) = 1 | + 1.30·2-s − 0.577·3-s + 0.714·4-s + 0.680·5-s − 0.755·6-s + 0.224·7-s − 0.374·8-s + 0.333·9-s + 0.891·10-s + 1.75·11-s − 0.412·12-s + 0.219·13-s + 0.293·14-s − 0.393·15-s − 1.20·16-s − 0.242·17-s + 0.436·18-s − 1.91·19-s + 0.486·20-s − 0.129·21-s + 2.29·22-s − 0.948·23-s + 0.215·24-s − 0.536·25-s + 0.287·26-s − 0.192·27-s + 0.160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 - 0.592T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 - 0.790T + 13T^{2} \) |
| 19 | \( 1 + 8.36T + 19T^{2} \) |
| 23 | \( 1 + 4.55T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 + 7.36T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 + 8.60T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 9.55T + 61T^{2} \) |
| 67 | \( 1 - 4.56T + 67T^{2} \) |
| 71 | \( 1 + 8.15T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−6.93078484961211569145947204514, −6.43021556155000504520038530684, −6.19626559564374902361383360718, −5.29717654716145391118039953420, −4.74253195725040011113532807795, −3.85529822310774103680931715713, −3.63034383669984234681046629621, −2.12514593659360542548873836224, −1.67267944772396336852959311434, 0,
1.67267944772396336852959311434, 2.12514593659360542548873836224, 3.63034383669984234681046629621, 3.85529822310774103680931715713, 4.74253195725040011113532807795, 5.29717654716145391118039953420, 6.19626559564374902361383360718, 6.43021556155000504520038530684, 6.93078484961211569145947204514