| L(s) = 1 | + 1.49·2-s + 0.225·4-s + 2.23·7-s − 2.64·8-s + 5.15·11-s + 1.45·13-s + 3.33·14-s − 4.40·16-s − 0.150·17-s − 4.63·19-s + 7.68·22-s + 1.12·23-s + 2.16·26-s + 0.504·28-s + 1.45·29-s + 6.63·31-s − 1.27·32-s − 0.224·34-s − 37-s − 6.91·38-s + 8.43·41-s + 0.405·43-s + 1.16·44-s + 1.68·46-s + 3.42·47-s − 2.01·49-s + 0.328·52-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 0.112·4-s + 0.843·7-s − 0.935·8-s + 1.55·11-s + 0.403·13-s + 0.890·14-s − 1.10·16-s − 0.0364·17-s − 1.06·19-s + 1.63·22-s + 0.235·23-s + 0.425·26-s + 0.0952·28-s + 0.270·29-s + 1.19·31-s − 0.224·32-s − 0.0384·34-s − 0.164·37-s − 1.12·38-s + 1.31·41-s + 0.0618·43-s + 0.175·44-s + 0.248·46-s + 0.500·47-s − 0.287·49-s + 0.0454·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.925390839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.925390839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 - 5.15T + 11T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 17 | \( 1 + 0.150T + 17T^{2} \) |
| 19 | \( 1 + 4.63T + 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 6.63T + 31T^{2} \) |
| 41 | \( 1 - 8.43T + 41T^{2} \) |
| 43 | \( 1 - 0.405T + 43T^{2} \) |
| 47 | \( 1 - 3.42T + 47T^{2} \) |
| 53 | \( 1 - 4.29T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 0.826T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 8.73T + 71T^{2} \) |
| 73 | \( 1 - 4.18T + 73T^{2} \) |
| 79 | \( 1 - 9.00T + 79T^{2} \) |
| 83 | \( 1 - 3.54T + 83T^{2} \) |
| 89 | \( 1 + 5.74T + 89T^{2} \) |
| 97 | \( 1 - 1.50T + 97T^{2} \) |
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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
−7.82334690919843337097939445770, −6.77245621369728202687450039455, −6.32130702798576004614642503993, −5.71182570163692895003681833836, −4.76105811728571865254415093412, −4.34258875138990253559066885488, −3.77027499131540210036880036001, −2.86562923284906388199224857609, −1.88106334510071070893749429698, −0.864643927530828103145479596813,
0.864643927530828103145479596813, 1.88106334510071070893749429698, 2.86562923284906388199224857609, 3.77027499131540210036880036001, 4.34258875138990253559066885488, 4.76105811728571865254415093412, 5.71182570163692895003681833836, 6.32130702798576004614642503993, 6.77245621369728202687450039455, 7.82334690919843337097939445770
