| L(s) = 1 | + 1.09·2-s − 0.800·4-s − 5-s + 0.705·7-s − 3.06·8-s − 1.09·10-s + 4.71·13-s + 0.772·14-s − 1.75·16-s − 7.78·17-s + 1.19·19-s + 0.800·20-s + 6.89·23-s + 25-s + 5.16·26-s − 0.564·28-s + 1.32·29-s − 7.68·31-s + 4.20·32-s − 8.52·34-s − 0.705·35-s + 8.43·37-s + 1.31·38-s + 3.06·40-s − 0.232·41-s − 7.32·43-s + 7.55·46-s + ⋯ |
| L(s) = 1 | + 0.774·2-s − 0.400·4-s − 0.447·5-s + 0.266·7-s − 1.08·8-s − 0.346·10-s + 1.30·13-s + 0.206·14-s − 0.439·16-s − 1.88·17-s + 0.275·19-s + 0.178·20-s + 1.43·23-s + 0.200·25-s + 1.01·26-s − 0.106·28-s + 0.246·29-s − 1.37·31-s + 0.743·32-s − 1.46·34-s − 0.119·35-s + 1.38·37-s + 0.213·38-s + 0.484·40-s − 0.0363·41-s − 1.11·43-s + 1.11·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 7 | \( 1 - 0.705T + 7T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 + 7.78T + 17T^{2} \) |
| 19 | \( 1 - 1.19T + 19T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 - 1.32T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 - 8.43T + 37T^{2} \) |
| 41 | \( 1 + 0.232T + 41T^{2} \) |
| 43 | \( 1 + 7.32T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 + 6.82T + 53T^{2} \) |
| 59 | \( 1 - 3.54T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 + 0.670T + 71T^{2} \) |
| 73 | \( 1 - 5.00T + 73T^{2} \) |
| 79 | \( 1 + 2.28T + 79T^{2} \) |
| 83 | \( 1 + 2.10T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 - 3.32T + 97T^{2} \) |
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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
−7.947802610125718165790168725695, −6.76583281404596437857930525759, −6.45819046555847937781604750403, −5.41233658576742079096486571861, −4.84229447624418319208764712899, −4.11839080617858791599722919806, −3.50255410938031383924822291401, −2.64616507775880874417059162614, −1.34257603859475631557253942658, 0,
1.34257603859475631557253942658, 2.64616507775880874417059162614, 3.50255410938031383924822291401, 4.11839080617858791599722919806, 4.84229447624418319208764712899, 5.41233658576742079096486571861, 6.45819046555847937781604750403, 6.76583281404596437857930525759, 7.947802610125718165790168725695
