| L(s) = 1 | + 2-s + 4-s − 1.47·7-s + 8-s + 2.29·11-s − 5.29·13-s − 1.47·14-s + 16-s + 1.47·17-s − 19-s + 2.29·22-s − 1.86·23-s − 5.29·26-s − 1.47·28-s − 7.16·29-s + 0.0470·31-s + 32-s + 1.47·34-s + 6.59·37-s − 38-s − 0.179·41-s + 7.98·43-s + 2.29·44-s − 1.86·46-s + 12.4·47-s − 4.82·49-s − 5.29·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.558·7-s + 0.353·8-s + 0.692·11-s − 1.46·13-s − 0.394·14-s + 0.250·16-s + 0.358·17-s − 0.229·19-s + 0.489·22-s − 0.389·23-s − 1.03·26-s − 0.279·28-s − 1.33·29-s + 0.00845·31-s + 0.176·32-s + 0.253·34-s + 1.08·37-s − 0.162·38-s − 0.0280·41-s + 1.21·43-s + 0.346·44-s − 0.275·46-s + 1.81·47-s − 0.688·49-s − 0.734·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 1.47T + 7T^{2} \) |
| 11 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + 7.16T + 29T^{2} \) |
| 31 | \( 1 - 0.0470T + 31T^{2} \) |
| 37 | \( 1 - 6.59T + 37T^{2} \) |
| 41 | \( 1 + 0.179T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 6.34T + 59T^{2} \) |
| 61 | \( 1 + 9.93T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 6.11T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−7.50205758102059509805304124407, −6.68863249250605218438765759024, −5.90693115706123916205285929007, −5.50420412663478528617957155201, −4.38674743721012348506771336811, −4.10979918409140993415180965066, −3.03129174503983615466834982550, −2.46255843408285878057994504854, −1.41367915655116801823959000825, 0,
1.41367915655116801823959000825, 2.46255843408285878057994504854, 3.03129174503983615466834982550, 4.10979918409140993415180965066, 4.38674743721012348506771336811, 5.50420412663478528617957155201, 5.90693115706123916205285929007, 6.68863249250605218438765759024, 7.50205758102059509805304124407
