| L(s) = 1 | − 2-s + 1.44·3-s + 4-s − 1.38·5-s − 1.44·6-s − 1.98·7-s − 8-s − 0.911·9-s + 1.38·10-s − 3.45·11-s + 1.44·12-s + 6.11·13-s + 1.98·14-s − 2.00·15-s + 16-s + 4.25·17-s + 0.911·18-s + 2.16·19-s − 1.38·20-s − 2.87·21-s + 3.45·22-s − 0.571·23-s − 1.44·24-s − 3.07·25-s − 6.11·26-s − 5.65·27-s − 1.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.834·3-s + 0.5·4-s − 0.619·5-s − 0.589·6-s − 0.750·7-s − 0.353·8-s − 0.303·9-s + 0.438·10-s − 1.04·11-s + 0.417·12-s + 1.69·13-s + 0.530·14-s − 0.517·15-s + 0.250·16-s + 1.03·17-s + 0.214·18-s + 0.497·19-s − 0.309·20-s − 0.626·21-s + 0.736·22-s − 0.119·23-s − 0.294·24-s − 0.615·25-s − 1.19·26-s − 1.08·27-s − 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243982177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243982177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 + 0.571T + 23T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 + 2.71T + 37T^{2} \) |
| 41 | \( 1 - 9.14T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 9.01T + 59T^{2} \) |
| 61 | \( 1 - 9.59T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 - 6.98T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 2.85T + 89T^{2} \) |
| 97 | \( 1 + 2.42T + 97T^{2} \) |
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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
−9.352687857403066054634428742250, −8.288582476086565539060259924485, −8.127417809694202835869764990483, −7.30309815910453893481844534271, −6.18170481389868619694086258023, −5.52171297262732123795638651908, −3.89891285299464191867315248915, −3.27639738589141332125048707356, −2.42091750775289024228045703886, −0.816413476773546080974234511699,
0.816413476773546080974234511699, 2.42091750775289024228045703886, 3.27639738589141332125048707356, 3.89891285299464191867315248915, 5.52171297262732123795638651908, 6.18170481389868619694086258023, 7.30309815910453893481844534271, 8.127417809694202835869764990483, 8.288582476086565539060259924485, 9.352687857403066054634428742250
