| L(s) = 1 | + 2-s + 2.74·3-s + 4-s − 5-s + 2.74·6-s − 3.29·7-s + 8-s + 4.53·9-s − 10-s − 11-s + 2.74·12-s + 0.737·13-s − 3.29·14-s − 2.74·15-s + 16-s + 5.18·17-s + 4.53·18-s − 2.64·19-s − 20-s − 9.03·21-s − 22-s + 4.87·23-s + 2.74·24-s + 25-s + 0.737·26-s + 4.21·27-s − 3.29·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.58·3-s + 0.5·4-s − 0.447·5-s + 1.12·6-s − 1.24·7-s + 0.353·8-s + 1.51·9-s − 0.316·10-s − 0.301·11-s + 0.792·12-s + 0.204·13-s − 0.879·14-s − 0.708·15-s + 0.250·16-s + 1.25·17-s + 1.06·18-s − 0.605·19-s − 0.223·20-s − 1.97·21-s − 0.213·22-s + 1.01·23-s + 0.560·24-s + 0.200·25-s + 0.144·26-s + 0.811·27-s − 0.621·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.076886691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.076886691\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 13 | \( 1 - 0.737T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 + 2.98T + 31T^{2} \) |
| 37 | \( 1 - 3.61T + 37T^{2} \) |
| 41 | \( 1 - 4.96T + 41T^{2} \) |
| 43 | \( 1 - 4.42T + 43T^{2} \) |
| 47 | \( 1 - 9.96T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 2.54T + 89T^{2} \) |
| 97 | \( 1 - 9.07T + 97T^{2} \) |
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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
−7.71914851568585169928865522815, −7.31606290826523513644699315957, −6.49459212252696063228049572610, −5.81090222203908766951507721487, −4.82260416116986659930430707340, −3.95645093360621607930121776771, −3.43579340758968200165158734954, −2.90565338993244157912852093259, −2.25506692710780585285903013999, −0.930444309976902578326486919704,
0.930444309976902578326486919704, 2.25506692710780585285903013999, 2.90565338993244157912852093259, 3.43579340758968200165158734954, 3.95645093360621607930121776771, 4.82260416116986659930430707340, 5.81090222203908766951507721487, 6.49459212252696063228049572610, 7.31606290826523513644699315957, 7.71914851568585169928865522815
