| L(s) = 1 | − 5-s + 4.27·7-s + 1.27·11-s + 6.27·13-s − 2·17-s + 19-s − 0.274·23-s + 25-s + 1.27·29-s − 1.27·31-s − 4.27·35-s + 4.54·37-s + 7.54·41-s + 4·43-s − 6.27·47-s + 11.2·49-s − 8.27·53-s − 1.27·55-s + 13·59-s − 6.54·61-s − 6.27·65-s − 14.5·67-s + 0.725·71-s − 15.0·73-s + 5.45·77-s − 4.54·79-s − 12.5·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.61·7-s + 0.384·11-s + 1.74·13-s − 0.485·17-s + 0.229·19-s − 0.0573·23-s + 0.200·25-s + 0.236·29-s − 0.228·31-s − 0.722·35-s + 0.747·37-s + 1.17·41-s + 0.609·43-s − 0.915·47-s + 1.61·49-s − 1.13·53-s − 0.171·55-s + 1.69·59-s − 0.838·61-s − 0.778·65-s − 1.77·67-s + 0.0860·71-s − 1.76·73-s + 0.621·77-s − 0.511·79-s − 1.37·83-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
| \(L(1)\) |
\(\approx\) |
\(2.408388056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408388056\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.27T + 7T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 - 6.27T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 0.274T + 23T^{2} \) |
| 29 | \( 1 - 1.27T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 - 7.54T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 6.27T + 47T^{2} \) |
| 53 | \( 1 + 8.27T + 53T^{2} \) |
| 59 | \( 1 - 13T + 59T^{2} \) |
| 61 | \( 1 + 6.54T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 0.725T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 9.82T + 89T^{2} \) |
| 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
−8.650252902838526726958790535983, −7.939727125040869020155615769110, −7.36933722094255852270549795771, −6.31209774848475950501927613175, −5.67184343606088208260550298262, −4.59101359525758670755932294012, −4.16850117286836626294597103096, −3.13152140550341023343367265983, −1.83263659250988989726856207869, −1.02789868228711771186798810681,
1.02789868228711771186798810681, 1.83263659250988989726856207869, 3.13152140550341023343367265983, 4.16850117286836626294597103096, 4.59101359525758670755932294012, 5.67184343606088208260550298262, 6.31209774848475950501927613175, 7.36933722094255852270549795771, 7.939727125040869020155615769110, 8.650252902838526726958790535983
