| L(s) = 1 | − 3-s + 5-s − 2.82·7-s + 9-s + 5.65·11-s − 13-s − 15-s − 4.82·17-s + 2.82·19-s + 2.82·21-s − 8.48·23-s + 25-s − 27-s − 3.17·29-s − 4·31-s − 5.65·33-s − 2.82·35-s − 0.343·37-s + 39-s + 3.65·41-s + 1.65·43-s + 45-s + 8·47-s + 1.00·49-s + 4.82·51-s − 9.31·53-s + 5.65·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.06·7-s + 0.333·9-s + 1.70·11-s − 0.277·13-s − 0.258·15-s − 1.17·17-s + 0.648·19-s + 0.617·21-s − 1.76·23-s + 0.200·25-s − 0.192·27-s − 0.588·29-s − 0.718·31-s − 0.984·33-s − 0.478·35-s − 0.0564·37-s + 0.160·39-s + 0.571·41-s + 0.252·43-s + 0.149·45-s + 1.16·47-s + 0.142·49-s + 0.676·51-s − 1.27·53-s + 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 0.343T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 2.48T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−8.464721825697258134528321243079, −7.25823228231439806205461535765, −6.71779660456640984196525117095, −6.09395214152480773155357457990, −5.49294305959171468014663005207, −4.21792458545098515994891300180, −3.78352541038029377804290039721, −2.49193862865707357080178719908, −1.42820444887224057751907431441, 0,
1.42820444887224057751907431441, 2.49193862865707357080178719908, 3.78352541038029377804290039721, 4.21792458545098515994891300180, 5.49294305959171468014663005207, 6.09395214152480773155357457990, 6.71779660456640984196525117095, 7.25823228231439806205461535765, 8.464721825697258134528321243079
