| L(s) = 1 | − 2-s + 2.30·3-s + 4-s + 5-s − 2.30·6-s + 3.30·7-s − 8-s + 2.30·9-s − 10-s − 2.60·11-s + 2.30·12-s − 2.30·13-s − 3.30·14-s + 2.30·15-s + 16-s + 1.30·17-s − 2.30·18-s − 0.605·19-s + 20-s + 7.60·21-s + 2.60·22-s − 6.90·23-s − 2.30·24-s + 25-s + 2.30·26-s − 1.60·27-s + 3.30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.447·5-s − 0.940·6-s + 1.24·7-s − 0.353·8-s + 0.767·9-s − 0.316·10-s − 0.785·11-s + 0.664·12-s − 0.638·13-s − 0.882·14-s + 0.594·15-s + 0.250·16-s + 0.315·17-s − 0.542·18-s − 0.138·19-s + 0.223·20-s + 1.65·21-s + 0.555·22-s − 1.44·23-s − 0.470·24-s + 0.200·25-s + 0.451·26-s − 0.308·27-s + 0.624·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570222813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570222813\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 + 0.605T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 + 0.605T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 - 3.30T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 7.69T + 59T^{2} \) |
| 61 | \( 1 - 0.302T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 8.90T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−11.67577167906280000360073252230, −10.57255252512302849701277700808, −9.769744200988371500358812565424, −8.855771223927276463027970058674, −7.945360191820628535753467075691, −7.59944135420357923935951520678, −5.89742003638108242312856480455, −4.50753762260778242446025361669, −2.80692499643066464749320708844, −1.85760263215716619713560207955,
1.85760263215716619713560207955, 2.80692499643066464749320708844, 4.50753762260778242446025361669, 5.89742003638108242312856480455, 7.59944135420357923935951520678, 7.945360191820628535753467075691, 8.855771223927276463027970058674, 9.769744200988371500358812565424, 10.57255252512302849701277700808, 11.67577167906280000360073252230
