| L(s) = 1 | − 2-s + 1.09·3-s + 4-s − 1.09·6-s + 3.20·7-s − 8-s − 1.80·9-s + 3.82·11-s + 1.09·12-s − 0.147·13-s − 3.20·14-s + 16-s − 0.978·17-s + 1.80·18-s + 2.67·19-s + 3.51·21-s − 3.82·22-s + 2.33·23-s − 1.09·24-s + 0.147·26-s − 5.25·27-s + 3.20·28-s + 6.30·29-s − 3.62·31-s − 32-s + 4.18·33-s + 0.978·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.631·3-s + 0.5·4-s − 0.446·6-s + 1.21·7-s − 0.353·8-s − 0.600·9-s + 1.15·11-s + 0.315·12-s − 0.0408·13-s − 0.857·14-s + 0.250·16-s − 0.237·17-s + 0.424·18-s + 0.613·19-s + 0.766·21-s − 0.815·22-s + 0.487·23-s − 0.223·24-s + 0.0288·26-s − 1.01·27-s + 0.606·28-s + 1.17·29-s − 0.650·31-s − 0.176·32-s + 0.728·33-s + 0.167·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874617049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874617049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 1.09T + 3T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 + 0.147T + 13T^{2} \) |
| 17 | \( 1 + 0.978T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 + 3.62T + 31T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 + 6.23T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 6.46T + 89T^{2} \) |
| 97 | \( 1 - 3.07T + 97T^{2} \) |
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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
−9.149632890408981723829804584040, −8.474766968965112134627029884307, −7.919864124275438965811064285926, −7.12060596099840916406405446605, −6.19492570500117007327145885226, −5.21555171778293336272398856608, −4.20008876152689217013182376358, −3.12615709160101338142673300366, −2.10666321650458873868823471400, −1.07584621765779447049463277129,
1.07584621765779447049463277129, 2.10666321650458873868823471400, 3.12615709160101338142673300366, 4.20008876152689217013182376358, 5.21555171778293336272398856608, 6.19492570500117007327145885226, 7.12060596099840916406405446605, 7.919864124275438965811064285926, 8.474766968965112134627029884307, 9.149632890408981723829804584040
