| L(s) = 1 | + 0.638·2-s − 1.59·4-s + 2.55·5-s − 2.98·7-s − 2.29·8-s + 1.63·10-s − 11-s − 1.61·13-s − 1.90·14-s + 1.71·16-s + 5.85·17-s + 0.125·19-s − 4.06·20-s − 0.638·22-s + 3.09·23-s + 1.51·25-s − 1.03·26-s + 4.74·28-s − 0.191·29-s − 1.81·31-s + 5.68·32-s + 3.74·34-s − 7.61·35-s − 5.11·37-s + 0.0804·38-s − 5.85·40-s − 6.37·41-s + ⋯ |
| L(s) = 1 | + 0.451·2-s − 0.795·4-s + 1.14·5-s − 1.12·7-s − 0.811·8-s + 0.515·10-s − 0.301·11-s − 0.447·13-s − 0.509·14-s + 0.429·16-s + 1.42·17-s + 0.0288·19-s − 0.908·20-s − 0.136·22-s + 0.645·23-s + 0.302·25-s − 0.202·26-s + 0.897·28-s − 0.0356·29-s − 0.326·31-s + 1.00·32-s + 0.641·34-s − 1.28·35-s − 0.841·37-s + 0.0130·38-s − 0.925·40-s − 0.995·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.638T + 2T^{2} \) |
| 5 | \( 1 - 2.55T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 19 | \( 1 - 0.125T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 0.191T + 29T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + 6.37T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 - 0.116T + 47T^{2} \) |
| 53 | \( 1 + 8.43T + 53T^{2} \) |
| 59 | \( 1 - 0.197T + 59T^{2} \) |
| 67 | \( 1 - 2.32T + 67T^{2} \) |
| 71 | \( 1 - 4.03T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 + 4.97T + 89T^{2} \) |
| 97 | \( 1 + 0.00505T + 97T^{2} \) |
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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
−7.72456327498639962628982367729, −6.78413233683164126464113903637, −6.17855223444760151714329244861, −5.34827978165808733408302997279, −5.19375595670182404037143582540, −3.95509326913224936039346521179, −3.25048530544763420914243529640, −2.58090701112526895706712583342, −1.30502716412589015946718347461, 0,
1.30502716412589015946718347461, 2.58090701112526895706712583342, 3.25048530544763420914243529640, 3.95509326913224936039346521179, 5.19375595670182404037143582540, 5.34827978165808733408302997279, 6.17855223444760151714329244861, 6.78413233683164126464113903637, 7.72456327498639962628982367729
