| L(s) = 1 | − 1.10·2-s + 0.955·3-s − 0.768·4-s − 3.55·5-s − 1.06·6-s + 7-s + 3.07·8-s − 2.08·9-s + 3.94·10-s − 4.00·11-s − 0.734·12-s − 1.10·14-s − 3.40·15-s − 1.87·16-s + 1.86·17-s + 2.31·18-s − 6.34·19-s + 2.73·20-s + 0.955·21-s + 4.44·22-s + 4.50·23-s + 2.93·24-s + 7.66·25-s − 4.86·27-s − 0.768·28-s + 8.63·29-s + 3.77·30-s + ⋯ |
| L(s) = 1 | − 0.784·2-s + 0.551·3-s − 0.384·4-s − 1.59·5-s − 0.433·6-s + 0.377·7-s + 1.08·8-s − 0.695·9-s + 1.24·10-s − 1.20·11-s − 0.211·12-s − 0.296·14-s − 0.878·15-s − 0.468·16-s + 0.452·17-s + 0.545·18-s − 1.45·19-s + 0.611·20-s + 0.208·21-s + 0.947·22-s + 0.938·23-s + 0.599·24-s + 1.53·25-s − 0.935·27-s − 0.145·28-s + 1.60·29-s + 0.689·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5489660808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5489660808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 3 | \( 1 - 0.955T + 3T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 - 4.50T + 23T^{2} \) |
| 29 | \( 1 - 8.63T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 0.832T + 47T^{2} \) |
| 53 | \( 1 - 6.53T + 53T^{2} \) |
| 59 | \( 1 - 5.20T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 - 3.05T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 0.0931T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−9.578144649462361518378609806118, −8.594382572170511898301255461944, −8.214549226854973824212512734985, −7.82186951292650378628565829596, −6.85648004096457691332727299730, −5.28842412273755994030363502192, −4.49597988639898134966314640835, −3.57309618453778161534560830933, −2.46492915988803384822535113554, −0.59069154560767254696380953613,
0.59069154560767254696380953613, 2.46492915988803384822535113554, 3.57309618453778161534560830933, 4.49597988639898134966314640835, 5.28842412273755994030363502192, 6.85648004096457691332727299730, 7.82186951292650378628565829596, 8.214549226854973824212512734985, 8.594382572170511898301255461944, 9.578144649462361518378609806118
