| L(s) = 1 | − 0.561·5-s + 11-s − 2·13-s − 7.12·17-s + 1.12·19-s + 7.68·23-s − 4.68·25-s + 7.12·29-s − 5.43·31-s + 5.68·37-s + 8.24·41-s + 1.12·43-s + 4·47-s − 7·49-s + 8.24·53-s − 0.561·55-s − 0.315·59-s − 9.36·61-s + 1.12·65-s − 7.68·67-s − 15.6·71-s − 6·73-s − 13.1·79-s − 11.3·83-s + 4·85-s − 0.561·89-s − 0.630·95-s + ⋯ |
| L(s) = 1 | − 0.251·5-s + 0.301·11-s − 0.554·13-s − 1.72·17-s + 0.257·19-s + 1.60·23-s − 0.936·25-s + 1.32·29-s − 0.976·31-s + 0.934·37-s + 1.28·41-s + 0.171·43-s + 0.583·47-s − 49-s + 1.13·53-s − 0.0757·55-s − 0.0410·59-s − 1.19·61-s + 0.139·65-s − 0.938·67-s − 1.86·71-s − 0.702·73-s − 1.47·79-s − 1.24·83-s + 0.433·85-s − 0.0595·89-s − 0.0647·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 5.43T + 31T^{2} \) |
| 37 | \( 1 - 5.68T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 0.315T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 0.561T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−7.48120377328319590204042855504, −7.12932146314020667769013688188, −6.29664385798322288985202319809, −5.59635727524482224712282294471, −4.53592211309218629690958572485, −4.29796067003829555892809938869, −3.07831325765273638898602444197, −2.42267594250387102154602829624, −1.27966770711495252836863617464, 0,
1.27966770711495252836863617464, 2.42267594250387102154602829624, 3.07831325765273638898602444197, 4.29796067003829555892809938869, 4.53592211309218629690958572485, 5.59635727524482224712282294471, 6.29664385798322288985202319809, 7.12932146314020667769013688188, 7.48120377328319590204042855504
