| L(s) = 1 | − 5-s − 3.27·7-s − 6.27·11-s − 1.27·13-s − 2·17-s + 19-s + 7.27·23-s + 25-s − 6.27·29-s + 6.27·31-s + 3.27·35-s − 10.5·37-s − 7.54·41-s + 4·43-s + 1.27·47-s + 3.72·49-s − 0.725·53-s + 6.27·55-s + 13·59-s + 8.54·61-s + 1.27·65-s + 0.549·67-s + 8.27·71-s + 15.0·73-s + 20.5·77-s + 10.5·79-s + 2.54·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.23·7-s − 1.89·11-s − 0.353·13-s − 0.485·17-s + 0.229·19-s + 1.51·23-s + 0.200·25-s − 1.16·29-s + 1.12·31-s + 0.553·35-s − 1.73·37-s − 1.17·41-s + 0.609·43-s + 0.185·47-s + 0.532·49-s − 0.0995·53-s + 0.846·55-s + 1.69·59-s + 1.09·61-s + 0.158·65-s + 0.0671·67-s + 0.982·71-s + 1.76·73-s + 2.34·77-s + 1.18·79-s + 0.279·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7653886278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7653886278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 3.27T + 7T^{2} \) |
| 11 | \( 1 + 6.27T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 7.27T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 7.54T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 1.27T + 47T^{2} \) |
| 53 | \( 1 + 0.725T + 53T^{2} \) |
| 59 | \( 1 - 13T + 59T^{2} \) |
| 61 | \( 1 - 8.54T + 61T^{2} \) |
| 67 | \( 1 - 0.549T + 67T^{2} \) |
| 71 | \( 1 - 8.27T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 2.54T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−8.597384557843943529786813673015, −7.913217828865460142882300000029, −7.05960089397053320680793935301, −6.64423680773547303557952435588, −5.37557109151116806806331633957, −5.06188546021354026268271835300, −3.78457887691571518364894280035, −3.04534604546798916282346167792, −2.30453229299001038824384109961, −0.49072456649946162630146174236,
0.49072456649946162630146174236, 2.30453229299001038824384109961, 3.04534604546798916282346167792, 3.78457887691571518364894280035, 5.06188546021354026268271835300, 5.37557109151116806806331633957, 6.64423680773547303557952435588, 7.05960089397053320680793935301, 7.913217828865460142882300000029, 8.597384557843943529786813673015
