| L(s) = 1 | + 2.64·2-s − 1.37·3-s + 4.97·4-s − 3.61·6-s − 3.64·7-s + 7.84·8-s − 1.12·9-s − 4.75·11-s − 6.81·12-s − 5.36·13-s − 9.62·14-s + 10.7·16-s + 1.33·17-s − 2.95·18-s + 4.99·21-s − 12.5·22-s + 5.06·23-s − 10.7·24-s − 14.1·26-s + 5.64·27-s − 18.1·28-s + 0.253·29-s + 5.11·31-s + 12.7·32-s + 6.52·33-s + 3.51·34-s − 5.57·36-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.791·3-s + 2.48·4-s − 1.47·6-s − 1.37·7-s + 2.77·8-s − 0.373·9-s − 1.43·11-s − 1.96·12-s − 1.48·13-s − 2.57·14-s + 2.69·16-s + 0.323·17-s − 0.697·18-s + 1.09·21-s − 2.67·22-s + 1.05·23-s − 2.19·24-s − 2.77·26-s + 1.08·27-s − 3.42·28-s + 0.0470·29-s + 0.918·31-s + 2.25·32-s + 1.13·33-s + 0.603·34-s − 0.928·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.196355487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.196355487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 3 | \( 1 + 1.37T + 3T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 - 0.253T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 9.97T + 41T^{2} \) |
| 43 | \( 1 - 1.99T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + 9.42T + 53T^{2} \) |
| 59 | \( 1 + 0.582T + 59T^{2} \) |
| 61 | \( 1 + 8.47T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 9.78T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 - 2.08T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 9.40T + 89T^{2} \) |
| 97 | \( 1 + 2.61T + 97T^{2} \) |
| 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
−7.42205734679178630516580421977, −6.70420141584432918035771537835, −6.04130436210955680242736299253, −5.71248519728767212594689752860, −4.83288781700304156250915111110, −4.63445532448650361933367018066, −3.40108222533609631100966996350, −2.72263785993726791962904478260, −2.50394017055604612679932512341, −0.62600297044355574591326549986,
0.62600297044355574591326549986, 2.50394017055604612679932512341, 2.72263785993726791962904478260, 3.40108222533609631100966996350, 4.63445532448650361933367018066, 4.83288781700304156250915111110, 5.71248519728767212594689752860, 6.04130436210955680242736299253, 6.70420141584432918035771537835, 7.42205734679178630516580421977
