| L(s) = 1 | − 0.514·2-s − 3-s − 1.73·4-s − 2.23·5-s + 0.514·6-s + 1.18·7-s + 1.92·8-s + 9-s + 1.14·10-s − 3.76·11-s + 1.73·12-s − 1.73·13-s − 0.611·14-s + 2.23·15-s + 2.48·16-s − 17-s − 0.514·18-s + 1.01·19-s + 3.87·20-s − 1.18·21-s + 1.93·22-s + 2.15·23-s − 1.92·24-s − 0.0244·25-s + 0.890·26-s − 27-s − 2.06·28-s + ⋯ |
| L(s) = 1 | − 0.363·2-s − 0.577·3-s − 0.867·4-s − 0.997·5-s + 0.209·6-s + 0.449·7-s + 0.679·8-s + 0.333·9-s + 0.362·10-s − 1.13·11-s + 0.501·12-s − 0.480·13-s − 0.163·14-s + 0.575·15-s + 0.620·16-s − 0.242·17-s − 0.121·18-s + 0.233·19-s + 0.865·20-s − 0.259·21-s + 0.412·22-s + 0.449·23-s − 0.392·24-s − 0.00488·25-s + 0.174·26-s − 0.192·27-s − 0.390·28-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 | \( 1 + 0.514T + 2T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 - 1.18T + 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 - 2.21T + 41T^{2} \) |
| 43 | \( 1 - 9.50T + 43T^{2} \) |
| 47 | \( 1 - 5.01T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 3.08T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 5.31T + 73T^{2} \) |
| 79 | \( 1 + 9.97T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 - 1.44T + 97T^{2} \) |
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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
−7.49219327273848737934682244506, −7.22982146230811808303541306661, −5.88754572556508276834784361537, −5.31983425086625004648577228500, −4.63825159719728571508274998514, −4.13441977460339870640589252968, −3.23966916393583817802406794566, −2.09956386686401479756470582376, −0.831861063167652053161786471456, 0,
0.831861063167652053161786471456, 2.09956386686401479756470582376, 3.23966916393583817802406794566, 4.13441977460339870640589252968, 4.63825159719728571508274998514, 5.31983425086625004648577228500, 5.88754572556508276834784361537, 7.22982146230811808303541306661, 7.49219327273848737934682244506
