| L(s) = 1 | − 1.27·2-s − 0.372·4-s + 2.15·5-s + 3.02·8-s − 2.74·10-s + 4.70·11-s − 2·13-s − 3.11·16-s + 2.15·17-s − 19-s − 0.800·20-s − 6·22-s + 6.85·23-s − 0.372·25-s + 2.55·26-s + 6.85·29-s + 6.74·31-s − 2.07·32-s − 2.74·34-s − 0.744·37-s + 1.27·38-s + 6.51·40-s − 2.55·41-s + 6.11·43-s − 1.75·44-s − 8.74·46-s + 9.00·47-s + ⋯ |
| L(s) = 1 | − 0.902·2-s − 0.186·4-s + 0.962·5-s + 1.07·8-s − 0.867·10-s + 1.41·11-s − 0.554·13-s − 0.779·16-s + 0.521·17-s − 0.229·19-s − 0.179·20-s − 1.27·22-s + 1.42·23-s − 0.0744·25-s + 0.500·26-s + 1.27·29-s + 1.21·31-s − 0.367·32-s − 0.470·34-s − 0.122·37-s + 0.206·38-s + 1.02·40-s − 0.398·41-s + 0.932·43-s − 0.263·44-s − 1.28·46-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.689710542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.689710542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 5 | \( 1 - 2.15T + 5T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 - 6.85T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + 0.744T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 5.10T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.88180463681743039961681981321, −7.18848532165089027617451062436, −6.50046678729917888145304610551, −5.90109879798660418430286739251, −4.88399620821069067192813995274, −4.46615241091671929845379170061, −3.40125030547526597947020145675, −2.42077029077740862072190833264, −1.45039929688945466590242236757, −0.829325516907455182392895298014,
0.829325516907455182392895298014, 1.45039929688945466590242236757, 2.42077029077740862072190833264, 3.40125030547526597947020145675, 4.46615241091671929845379170061, 4.88399620821069067192813995274, 5.90109879798660418430286739251, 6.50046678729917888145304610551, 7.18848532165089027617451062436, 7.88180463681743039961681981321
