| L(s) = 1 | − 2-s + 4-s − 5-s + 3.73·7-s − 8-s + 10-s + 3.46·11-s + 3.73·13-s − 3.73·14-s + 16-s − 3.46·17-s − 7.92·19-s − 20-s − 3.46·22-s − 0.464·23-s + 25-s − 3.73·26-s + 3.73·28-s + 6.92·29-s + 5.46·31-s − 32-s + 3.46·34-s − 3.73·35-s + 8·37-s + 7.92·38-s + 40-s − 5.19·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.41·7-s − 0.353·8-s + 0.316·10-s + 1.04·11-s + 1.03·13-s − 0.997·14-s + 0.250·16-s − 0.840·17-s − 1.81·19-s − 0.223·20-s − 0.738·22-s − 0.0967·23-s + 0.200·25-s − 0.731·26-s + 0.705·28-s + 1.28·29-s + 0.981·31-s − 0.176·32-s + 0.594·34-s − 0.630·35-s + 1.31·37-s + 1.28·38-s + 0.158·40-s − 0.811·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.284239900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284239900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 + 0.464T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 - 6.46T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 8.66T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 - 6.39T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−10.45460076605620586791405480300, −9.138612047685520954173492878430, −8.441863543464289910519136685131, −8.077706638150060410560643999503, −6.79199844504357611760716941557, −6.18349324003908300492456465149, −4.66769527117573781297638060484, −3.97253044885645380423220019559, −2.29648142144727740264232806462, −1.11968650787317546281280170299,
1.11968650787317546281280170299, 2.29648142144727740264232806462, 3.97253044885645380423220019559, 4.66769527117573781297638060484, 6.18349324003908300492456465149, 6.79199844504357611760716941557, 8.077706638150060410560643999503, 8.441863543464289910519136685131, 9.138612047685520954173492878430, 10.45460076605620586791405480300
