| L(s) = 1 | + 0.488·2-s − 3-s − 1.76·4-s − 0.529·5-s − 0.488·6-s − 1.83·8-s + 9-s − 0.258·10-s − 1.22·11-s + 1.76·12-s − 1.15·13-s + 0.529·15-s + 2.62·16-s + 6.79·17-s + 0.488·18-s − 2.34·19-s + 0.933·20-s − 0.599·22-s + 23-s + 1.83·24-s − 4.71·25-s − 0.561·26-s − 27-s − 5.30·29-s + 0.258·30-s − 10.3·31-s + 4.95·32-s + ⋯ |
| L(s) = 1 | + 0.345·2-s − 0.577·3-s − 0.880·4-s − 0.236·5-s − 0.199·6-s − 0.649·8-s + 0.333·9-s − 0.0817·10-s − 0.370·11-s + 0.508·12-s − 0.319·13-s + 0.136·15-s + 0.656·16-s + 1.64·17-s + 0.115·18-s − 0.537·19-s + 0.208·20-s − 0.127·22-s + 0.208·23-s + 0.374·24-s − 0.943·25-s − 0.110·26-s − 0.192·27-s − 0.984·29-s + 0.0472·30-s − 1.85·31-s + 0.875·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9075992966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9075992966\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.488T + 2T^{2} \) |
| 5 | \( 1 + 0.529T + 5T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 7.49T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 7.21T + 89T^{2} \) |
| 97 | \( 1 - 8.00T + 97T^{2} \) |
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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
−8.541840794593517566583912632813, −7.85277028875415549731563447818, −7.18666612669471321353599382323, −6.11888294709081551053447667476, −5.36358319957575634273451914612, −5.03850209642571581552222326837, −3.84289280642658409561495990463, −3.46192783360356546227362096741, −1.96771499937637682527284007497, −0.54834769413819439096676485530,
0.54834769413819439096676485530, 1.96771499937637682527284007497, 3.46192783360356546227362096741, 3.84289280642658409561495990463, 5.03850209642571581552222326837, 5.36358319957575634273451914612, 6.11888294709081551053447667476, 7.18666612669471321353599382323, 7.85277028875415549731563447818, 8.541840794593517566583912632813
