| L(s) = 1 | + 1.45·2-s + 0.117·4-s + 3.93·5-s − 7-s − 2.73·8-s + 5.73·10-s − 0.418·11-s + 5.47·13-s − 1.45·14-s − 4.22·16-s + 17-s + 3.77·19-s + 0.463·20-s − 0.608·22-s + 2.88·23-s + 10.5·25-s + 7.96·26-s − 0.117·28-s − 3.24·29-s + 2.82·31-s − 0.665·32-s + 1.45·34-s − 3.93·35-s − 9.10·37-s + 5.49·38-s − 10.7·40-s + 11.2·41-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.0588·4-s + 1.76·5-s − 0.377·7-s − 0.968·8-s + 1.81·10-s − 0.126·11-s + 1.51·13-s − 0.388·14-s − 1.05·16-s + 0.242·17-s + 0.866·19-s + 0.103·20-s − 0.129·22-s + 0.600·23-s + 2.10·25-s + 1.56·26-s − 0.0222·28-s − 0.603·29-s + 0.506·31-s − 0.117·32-s + 0.249·34-s − 0.665·35-s − 1.49·37-s + 0.891·38-s − 1.70·40-s + 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.268268519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.268268519\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 1.45T + 2T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 11 | \( 1 + 0.418T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 19 | \( 1 - 3.77T + 19T^{2} \) |
| 23 | \( 1 - 2.88T + 23T^{2} \) |
| 29 | \( 1 + 3.24T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 9.10T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 7.80T + 47T^{2} \) |
| 53 | \( 1 + 4.03T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 - 8.55T + 73T^{2} \) |
| 79 | \( 1 + 2.59T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 5.72T + 89T^{2} \) |
| 97 | \( 1 + 2.07T + 97T^{2} \) |
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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
−9.792570066493311929042744776775, −9.208439273710540615555631581542, −8.478874147126600632310264585290, −6.95969428689944332493365356261, −6.09802372435486760585781617472, −5.67590830614464609989610706336, −4.86829673925598306035700163748, −3.59385269883443013112759973586, −2.78325600122807694682718545314, −1.40141601568346004779122391158,
1.40141601568346004779122391158, 2.78325600122807694682718545314, 3.59385269883443013112759973586, 4.86829673925598306035700163748, 5.67590830614464609989610706336, 6.09802372435486760585781617472, 6.95969428689944332493365356261, 8.478874147126600632310264585290, 9.208439273710540615555631581542, 9.792570066493311929042744776775
