| 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
| \(L(1)\) |
\(\approx\) |
\(1.876802205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876802205\) |
| \(L(\frac{3}{2})\) |
|
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
−9.331237810399224392101313805354, −8.534203655761091443459072228367, −7.33840595176009413115567876453, −6.59496428328530600873352682895, −6.02158076808438538469207291472, −5.38013866605766473665492678496, −4.22339937749192440247014231788, −3.45749263282355228572280727513, −2.53094124719999746131879283368, −0.865789801876275995475213352872,
0.865789801876275995475213352872, 2.53094124719999746131879283368, 3.45749263282355228572280727513, 4.22339937749192440247014231788, 5.38013866605766473665492678496, 6.02158076808438538469207291472, 6.59496428328530600873352682895, 7.33840595176009413115567876453, 8.534203655761091443459072228367, 9.331237810399224392101313805354
