| L(s) = 1 | − 3.23·5-s + 2·7-s − 3.23·11-s − 4.47·13-s + 6.47·17-s − 1.23·19-s − 23-s + 5.47·25-s + 8.47·29-s + 1.52·31-s − 6.47·35-s + 7.70·37-s + 2·41-s + 5.23·43-s + 8.94·47-s − 3·49-s − 12.1·53-s + 10.4·55-s + 4·59-s − 11.7·61-s + 14.4·65-s + 7.70·67-s − 8.94·71-s + 4.47·73-s − 6.47·77-s + 14·79-s + 11.2·83-s + ⋯ |
| L(s) = 1 | − 1.44·5-s + 0.755·7-s − 0.975·11-s − 1.24·13-s + 1.56·17-s − 0.283·19-s − 0.208·23-s + 1.09·25-s + 1.57·29-s + 0.274·31-s − 1.09·35-s + 1.26·37-s + 0.312·41-s + 0.798·43-s + 1.30·47-s − 0.428·49-s − 1.67·53-s + 1.41·55-s + 0.520·59-s − 1.49·61-s + 1.79·65-s + 0.941·67-s − 1.06·71-s + 0.523·73-s − 0.737·77-s + 1.57·79-s + 1.23·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.172318069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172318069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−9.366846903582688002131083226352, −8.178301476670489846119572787890, −7.82346278649890985025887217592, −7.39579432157154462655075203528, −6.11574687058138252639114592336, −4.95787379854127402370668369715, −4.54696459059183604183769577140, −3.38677469240677652336054206190, −2.45625249591834147116591770369, −0.74405639684822021400048784867,
0.74405639684822021400048784867, 2.45625249591834147116591770369, 3.38677469240677652336054206190, 4.54696459059183604183769577140, 4.95787379854127402370668369715, 6.11574687058138252639114592336, 7.39579432157154462655075203528, 7.82346278649890985025887217592, 8.178301476670489846119572787890, 9.366846903582688002131083226352
