| L(s) = 1 | − 0.655·2-s − 1.57·4-s + 5-s − 0.777·7-s + 2.33·8-s − 0.655·10-s + 4.06·11-s − 13-s + 0.509·14-s + 1.60·16-s − 5.29·17-s + 6.65·19-s − 1.57·20-s − 2.66·22-s − 2.30·23-s + 25-s + 0.655·26-s + 1.22·28-s − 6.38·29-s − 8.94·31-s − 5.73·32-s + 3.46·34-s − 0.777·35-s − 2.80·37-s − 4.35·38-s + 2.33·40-s − 7.60·41-s + ⋯ |
| L(s) = 1 | − 0.463·2-s − 0.785·4-s + 0.447·5-s − 0.293·7-s + 0.827·8-s − 0.207·10-s + 1.22·11-s − 0.277·13-s + 0.136·14-s + 0.402·16-s − 1.28·17-s + 1.52·19-s − 0.351·20-s − 0.568·22-s − 0.480·23-s + 0.200·25-s + 0.128·26-s + 0.230·28-s − 1.18·29-s − 1.60·31-s − 1.01·32-s + 0.594·34-s − 0.131·35-s − 0.461·37-s − 0.706·38-s + 0.369·40-s − 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.655T + 2T^{2} \) |
| 7 | \( 1 + 0.777T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 + 6.38T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 2.80T + 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 - 4.74T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 0.307T + 71T^{2} \) |
| 73 | \( 1 + 7.50T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 6.08T + 97T^{2} \) |
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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
−7.82608862792584976499988458389, −7.22675488212863359215643759944, −6.48956420416262400430406837355, −5.59985064406650826428229175287, −4.98134875587825957858792664820, −4.00832861601861367145107329741, −3.49211576577292451680866562934, −2.12667362889965779960754371801, −1.26721484264216815446746383889, 0,
1.26721484264216815446746383889, 2.12667362889965779960754371801, 3.49211576577292451680866562934, 4.00832861601861367145107329741, 4.98134875587825957858792664820, 5.59985064406650826428229175287, 6.48956420416262400430406837355, 7.22675488212863359215643759944, 7.82608862792584976499988458389
