| L(s) = 1 | + 2-s + 3-s + 4-s + 2.71·5-s + 6-s + 1.56·7-s + 8-s + 9-s + 2.71·10-s + 3.34·11-s + 12-s − 0.269·13-s + 1.56·14-s + 2.71·15-s + 16-s + 17-s + 18-s + 4.26·19-s + 2.71·20-s + 1.56·21-s + 3.34·22-s + 8.99·23-s + 24-s + 2.39·25-s − 0.269·26-s + 27-s + 1.56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.21·5-s + 0.408·6-s + 0.593·7-s + 0.353·8-s + 0.333·9-s + 0.859·10-s + 1.00·11-s + 0.288·12-s − 0.0747·13-s + 0.419·14-s + 0.701·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.977·19-s + 0.607·20-s + 0.342·21-s + 0.713·22-s + 1.87·23-s + 0.204·24-s + 0.478·25-s − 0.0528·26-s + 0.192·27-s + 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.244561571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.244561571\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 + 0.269T + 13T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 23 | \( 1 - 8.99T + 23T^{2} \) |
| 29 | \( 1 + 8.55T + 29T^{2} \) |
| 31 | \( 1 - 5.67T + 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 61 | \( 1 + 6.19T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 6.38T + 73T^{2} \) |
| 79 | \( 1 - 1.50T + 79T^{2} \) |
| 83 | \( 1 + 6.12T + 83T^{2} \) |
| 89 | \( 1 - 7.05T + 89T^{2} \) |
| 97 | \( 1 - 4.17T + 97T^{2} \) |
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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
−8.008974710868917610333760151043, −7.22837665310960762924981539055, −6.60984669512052468743305371061, −5.89311861990183948466049943186, −5.08575067699832603684968080760, −4.63693951911240289697064593116, −3.43335918777806996280361402836, −2.98433004887326116868840384639, −1.74323878198127066845153166555, −1.41723834601324024683781150330,
1.41723834601324024683781150330, 1.74323878198127066845153166555, 2.98433004887326116868840384639, 3.43335918777806996280361402836, 4.63693951911240289697064593116, 5.08575067699832603684968080760, 5.89311861990183948466049943186, 6.60984669512052468743305371061, 7.22837665310960762924981539055, 8.008974710868917610333760151043
