| L(s) = 1 | − 0.921·2-s − 1.15·4-s − 5-s + 3.92·7-s + 2.90·8-s + 0.921·10-s − 5.05·11-s − 0.610·13-s − 3.61·14-s − 0.371·16-s + 2.84·17-s − 3.27·19-s + 1.15·20-s + 4.65·22-s − 4.90·23-s + 25-s + 0.562·26-s − 4.51·28-s + 4.76·29-s + 3.72·31-s − 5.46·32-s − 2.62·34-s − 3.92·35-s + 3.36·37-s + 3.01·38-s − 2.90·40-s − 0.562·41-s + ⋯ |
| L(s) = 1 | − 0.651·2-s − 0.575·4-s − 0.447·5-s + 1.48·7-s + 1.02·8-s + 0.291·10-s − 1.52·11-s − 0.169·13-s − 0.965·14-s − 0.0928·16-s + 0.690·17-s − 0.750·19-s + 0.257·20-s + 0.993·22-s − 1.02·23-s + 0.200·25-s + 0.110·26-s − 0.853·28-s + 0.885·29-s + 0.669·31-s − 0.965·32-s − 0.449·34-s − 0.662·35-s + 0.553·37-s + 0.489·38-s − 0.459·40-s − 0.0878·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9438670047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9438670047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 + 0.921T + 2T^{2} \) |
| 7 | \( 1 - 3.92T + 7T^{2} \) |
| 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 + 0.610T + 13T^{2} \) |
| 17 | \( 1 - 2.84T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 - 3.72T + 31T^{2} \) |
| 37 | \( 1 - 3.36T + 37T^{2} \) |
| 41 | \( 1 + 0.562T + 41T^{2} \) |
| 43 | \( 1 + 2.21T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 8.43T + 53T^{2} \) |
| 59 | \( 1 - 8.67T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 + 8.41T + 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 - 5.45T + 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 0.141T + 89T^{2} \) |
| 97 | \( 1 - 9.79T + 97T^{2} \) |
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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
−8.298038730449026264177105961342, −7.81168420591118434233540928574, −7.29218403586290091061214374245, −6.00614621382071491498671880721, −5.11214727741706987391203957183, −4.69252301342098445583897087273, −3.95514819475150774915862898287, −2.68846795866958774730978688809, −1.73275673018029055470973764045, −0.60564849977245159932583855442,
0.60564849977245159932583855442, 1.73275673018029055470973764045, 2.68846795866958774730978688809, 3.95514819475150774915862898287, 4.69252301342098445583897087273, 5.11214727741706987391203957183, 6.00614621382071491498671880721, 7.29218403586290091061214374245, 7.81168420591118434233540928574, 8.298038730449026264177105961342
