| L(s) = 1 | + 1.36·2-s − 0.146·3-s − 0.134·4-s + 1.91·5-s − 0.200·6-s − 1.87·7-s − 2.91·8-s − 2.97·9-s + 2.62·10-s − 11-s + 0.0197·12-s − 0.0301·13-s − 2.56·14-s − 0.281·15-s − 3.71·16-s − 17-s − 4.06·18-s + 1.35·19-s − 0.257·20-s + 0.275·21-s − 1.36·22-s − 5.38·23-s + 0.428·24-s − 1.31·25-s − 0.0412·26-s + 0.878·27-s + 0.251·28-s + ⋯ |
| L(s) = 1 | + 0.965·2-s − 0.0848·3-s − 0.0670·4-s + 0.857·5-s − 0.0819·6-s − 0.709·7-s − 1.03·8-s − 0.992·9-s + 0.828·10-s − 0.301·11-s + 0.00568·12-s − 0.00836·13-s − 0.685·14-s − 0.0727·15-s − 0.928·16-s − 0.242·17-s − 0.958·18-s + 0.310·19-s − 0.0575·20-s + 0.0602·21-s − 0.291·22-s − 1.12·23-s + 0.0874·24-s − 0.263·25-s − 0.00808·26-s + 0.169·27-s + 0.0475·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.010112570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.010112570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 3 | \( 1 + 0.146T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 13 | \( 1 + 0.0301T + 13T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 - 2.15T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 0.880T + 41T^{2} \) |
| 47 | \( 1 - 4.75T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 5.77T + 59T^{2} \) |
| 61 | \( 1 + 8.38T + 61T^{2} \) |
| 67 | \( 1 - 2.60T + 67T^{2} \) |
| 71 | \( 1 - 0.977T + 71T^{2} \) |
| 73 | \( 1 - 9.22T + 73T^{2} \) |
| 79 | \( 1 + 0.903T + 79T^{2} \) |
| 83 | \( 1 - 3.64T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.927986903984455537794951840586, −6.74859531133483042706443476147, −6.06619527932475083983065890604, −5.88999089175951574100564865518, −5.07649134209743459592541597705, −4.37724106452650949229040383038, −3.49410895547232561857199321268, −2.80798832349772610783335370654, −2.16765726026649663458002120042, −0.57546041515355145202725250986,
0.57546041515355145202725250986, 2.16765726026649663458002120042, 2.80798832349772610783335370654, 3.49410895547232561857199321268, 4.37724106452650949229040383038, 5.07649134209743459592541597705, 5.88999089175951574100564865518, 6.06619527932475083983065890604, 6.74859531133483042706443476147, 7.927986903984455537794951840586
