| L(s) = 1 | − 28.9·3-s − 146.·7-s + 594.·9-s − 191.·11-s + 83.9·13-s − 2.00e3·17-s − 677.·19-s + 4.24e3·21-s − 1.29e3·23-s − 1.01e4·27-s − 3.26e3·29-s − 6.15e3·31-s + 5.53e3·33-s − 1.13e4·37-s − 2.43e3·39-s − 1.05e4·41-s − 1.29e4·43-s − 9.52e3·47-s + 4.75e3·49-s + 5.78e4·51-s + 1.47e4·53-s + 1.96e4·57-s + 3.82e4·59-s − 3.58e3·61-s − 8.72e4·63-s − 2.17e4·67-s + 3.75e4·69-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 1.13·7-s + 2.44·9-s − 0.476·11-s + 0.137·13-s − 1.67·17-s − 0.430·19-s + 2.10·21-s − 0.510·23-s − 2.68·27-s − 0.721·29-s − 1.15·31-s + 0.883·33-s − 1.36·37-s − 0.255·39-s − 0.984·41-s − 1.06·43-s − 0.628·47-s + 0.282·49-s + 3.11·51-s + 0.723·53-s + 0.799·57-s + 1.42·59-s − 0.123·61-s − 2.76·63-s − 0.592·67-s + 0.948·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.06873047032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06873047032\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 28.9T + 243T^{2} \) |
| 7 | \( 1 + 146.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 191.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 83.9T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.00e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 677.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.52e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.08e4T + 8.58e9T^{2} \) |
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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
−10.55395317395842073112985598598, −9.917304526605672854681571208305, −8.773284183234977834704710605394, −7.15543090134887788515540022284, −6.57571686656275759263355047591, −5.75765811202711115887866703842, −4.82752737746971105755635848255, −3.70249224154166266541178376348, −1.87233767178290343093012978841, −0.14597691267082802807982601729,
0.14597691267082802807982601729, 1.87233767178290343093012978841, 3.70249224154166266541178376348, 4.82752737746971105755635848255, 5.75765811202711115887866703842, 6.57571686656275759263355047591, 7.15543090134887788515540022284, 8.773284183234977834704710605394, 9.917304526605672854681571208305, 10.55395317395842073112985598598
