| L(s) = 1 | + 2-s + 4-s − 3.85·5-s + 7-s + 8-s − 3.85·10-s − 2.38·11-s + 0.763·13-s + 14-s + 16-s + 6.47·17-s − 19-s − 3.85·20-s − 2.38·22-s − 7.70·23-s + 9.85·25-s + 0.763·26-s + 28-s + 2.14·29-s + 8.47·31-s + 32-s + 6.47·34-s − 3.85·35-s − 1.85·37-s − 38-s − 3.85·40-s + 10.3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.72·5-s + 0.377·7-s + 0.353·8-s − 1.21·10-s − 0.718·11-s + 0.211·13-s + 0.267·14-s + 0.250·16-s + 1.56·17-s − 0.229·19-s − 0.861·20-s − 0.507·22-s − 1.60·23-s + 1.97·25-s + 0.149·26-s + 0.188·28-s + 0.398·29-s + 1.52·31-s + 0.176·32-s + 1.10·34-s − 0.651·35-s − 0.304·37-s − 0.162·38-s − 0.609·40-s + 1.61·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021035099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021035099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.85T + 5T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 6.94T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−8.572417015715651033150574995106, −7.943267884736368674291763708879, −7.66560285481341894954541045845, −6.66757177527337767734633191359, −5.69059916047118335369982586535, −4.86345975302589407860331433156, −4.06289763526495944216964580084, −3.47280229940645882563311363422, −2.45003777023784668631357170395, −0.822128709153847757450929443586,
0.822128709153847757450929443586, 2.45003777023784668631357170395, 3.47280229940645882563311363422, 4.06289763526495944216964580084, 4.86345975302589407860331433156, 5.69059916047118335369982586535, 6.66757177527337767734633191359, 7.66560285481341894954541045845, 7.943267884736368674291763708879, 8.572417015715651033150574995106
