| L(s) = 1 | + 2-s − 2.98·3-s + 4-s + 2.64·5-s − 2.98·6-s + 8-s + 5.90·9-s + 2.64·10-s − 11-s − 2.98·12-s + 1.74·13-s − 7.88·15-s + 16-s − 0.301·17-s + 5.90·18-s − 0.384·19-s + 2.64·20-s − 22-s − 0.900·23-s − 2.98·24-s + 1.98·25-s + 1.74·26-s − 8.65·27-s + 4.98·29-s − 7.88·30-s + 8.26·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.72·3-s + 0.5·4-s + 1.18·5-s − 1.21·6-s + 0.353·8-s + 1.96·9-s + 0.835·10-s − 0.301·11-s − 0.861·12-s + 0.483·13-s − 2.03·15-s + 0.250·16-s − 0.0732·17-s + 1.39·18-s − 0.0882·19-s + 0.590·20-s − 0.213·22-s − 0.187·23-s − 0.608·24-s + 0.396·25-s + 0.341·26-s − 1.66·27-s + 0.925·29-s − 1.43·30-s + 1.48·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.781121986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781121986\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 0.301T + 17T^{2} \) |
| 19 | \( 1 + 0.384T + 19T^{2} \) |
| 23 | \( 1 + 0.900T + 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 0.559T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 47 | \( 1 - 3.66T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 - 8.57T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 6.31T + 71T^{2} \) |
| 73 | \( 1 - 3.96T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 3.36T + 89T^{2} \) |
| 97 | \( 1 + 9.47T + 97T^{2} \) |
| 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
−10.28137930606893507771166392903, −9.662044194837092820470332087017, −8.268808446566147828521641483003, −6.93448811898604778216454307108, −6.33162099871312052387698294493, −5.69761467768159272924371439715, −5.05899400742353649670720134080, −4.12177754834751421056024318050, −2.46748021447447705635954753526, −1.10594624941386690552921547358,
1.10594624941386690552921547358, 2.46748021447447705635954753526, 4.12177754834751421056024318050, 5.05899400742353649670720134080, 5.69761467768159272924371439715, 6.33162099871312052387698294493, 6.93448811898604778216454307108, 8.268808446566147828521641483003, 9.662044194837092820470332087017, 10.28137930606893507771166392903
