| L(s) = 1 | + 2.54·2-s − 2.46·3-s + 4.48·4-s − 6.26·6-s − 7-s + 6.31·8-s + 3.06·9-s − 4.70·11-s − 11.0·12-s + 2.32·13-s − 2.54·14-s + 7.11·16-s + 1.82·17-s + 7.80·18-s + 7.09·19-s + 2.46·21-s − 11.9·22-s + 23-s − 15.5·24-s + 5.92·26-s − 0.159·27-s − 4.48·28-s − 9.98·29-s + 3.53·31-s + 5.48·32-s + 11.5·33-s + 4.64·34-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 1.42·3-s + 2.24·4-s − 2.55·6-s − 0.377·7-s + 2.23·8-s + 1.02·9-s − 1.41·11-s − 3.18·12-s + 0.645·13-s − 0.680·14-s + 1.77·16-s + 0.443·17-s + 1.83·18-s + 1.62·19-s + 0.537·21-s − 2.55·22-s + 0.208·23-s − 3.17·24-s + 1.16·26-s − 0.0307·27-s − 0.846·28-s − 1.85·29-s + 0.635·31-s + 0.969·32-s + 2.01·33-s + 0.797·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.394347366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.394347366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 3 | \( 1 + 2.46T + 3T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 29 | \( 1 + 9.98T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 - 0.166T + 37T^{2} \) |
| 41 | \( 1 - 7.25T + 41T^{2} \) |
| 43 | \( 1 - 9.57T + 43T^{2} \) |
| 47 | \( 1 + 4.66T + 47T^{2} \) |
| 53 | \( 1 - 0.961T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 0.954T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 - 7.59T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 - 5.57T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−7.979530895472787558413998424473, −7.31570602352334414295290647492, −6.63647011268521973967109058209, −5.71987581058712131890358150037, −5.55147688013980852460921687779, −4.97833642844125336051173883878, −3.99592888680963863602200642469, −3.23888884765524645645014267858, −2.33091576500480927815931154436, −0.861909461123922764920358436227,
0.861909461123922764920358436227, 2.33091576500480927815931154436, 3.23888884765524645645014267858, 3.99592888680963863602200642469, 4.97833642844125336051173883878, 5.55147688013980852460921687779, 5.71987581058712131890358150037, 6.63647011268521973967109058209, 7.31570602352334414295290647492, 7.979530895472787558413998424473
