| L(s) = 1 | + 3·3-s − 4·5-s + 6·9-s + 2·11-s + 5·13-s − 12·15-s + 4·19-s − 23-s + 11·25-s + 9·27-s + 3·29-s + 5·31-s + 6·33-s − 4·37-s + 15·39-s − 5·41-s − 4·43-s − 24·45-s − 11·47-s − 8·55-s + 12·57-s + 12·59-s − 6·61-s − 20·65-s − 16·67-s − 3·69-s − 7·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s + 2·9-s + 0.603·11-s + 1.38·13-s − 3.09·15-s + 0.917·19-s − 0.208·23-s + 11/5·25-s + 1.73·27-s + 0.557·29-s + 0.898·31-s + 1.04·33-s − 0.657·37-s + 2.40·39-s − 0.780·41-s − 0.609·43-s − 3.57·45-s − 1.60·47-s − 1.07·55-s + 1.58·57-s + 1.56·59-s − 0.768·61-s − 2.48·65-s − 1.95·67-s − 0.361·69-s − 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.45167477185077, −13.87588642247854, −13.40775468546270, −13.09768774372747, −12.30446474695976, −11.75133340957886, −11.61602244672617, −10.85413661566642, −10.19133786222713, −9.788042956126378, −8.919026437166868, −8.630175699100555, −8.399169257069033, −7.735487868619013, −7.448474642460499, −6.763037929594324, −6.342259214573924, −5.266919412553640, −4.508222150791160, −4.071611645169539, −3.602737557356563, −3.142753030148152, −2.759667572399856, −1.495789390377401, −1.239132038030666, 0,
1.239132038030666, 1.495789390377401, 2.759667572399856, 3.142753030148152, 3.602737557356563, 4.071611645169539, 4.508222150791160, 5.266919412553640, 6.342259214573924, 6.763037929594324, 7.448474642460499, 7.735487868619013, 8.399169257069033, 8.630175699100555, 8.919026437166868, 9.788042956126378, 10.19133786222713, 10.85413661566642, 11.61602244672617, 11.75133340957886, 12.30446474695976, 13.09768774372747, 13.40775468546270, 13.87588642247854, 14.45167477185077
