| L(s) = 1 | − 1.85·2-s − 0.499·3-s + 1.42·4-s + 1.20·5-s + 0.923·6-s + 3.73·7-s + 1.06·8-s − 2.75·9-s − 2.23·10-s + 1.85·11-s − 0.710·12-s + 5.18·13-s − 6.90·14-s − 0.602·15-s − 4.82·16-s + 5.66·17-s + 5.08·18-s − 1.43·19-s + 1.71·20-s − 1.86·21-s − 3.43·22-s + 3.54·23-s − 0.532·24-s − 3.54·25-s − 9.59·26-s + 2.87·27-s + 5.31·28-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 0.288·3-s + 0.711·4-s + 0.539·5-s + 0.377·6-s + 1.41·7-s + 0.377·8-s − 0.916·9-s − 0.706·10-s + 0.559·11-s − 0.205·12-s + 1.43·13-s − 1.84·14-s − 0.155·15-s − 1.20·16-s + 1.37·17-s + 1.19·18-s − 0.330·19-s + 0.384·20-s − 0.406·21-s − 0.732·22-s + 0.739·23-s − 0.108·24-s − 0.708·25-s − 1.88·26-s + 0.552·27-s + 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9810142729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9810142729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 3 | \( 1 + 0.499T + 3T^{2} \) |
| 5 | \( 1 - 1.20T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 + 1.43T + 19T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 37 | \( 1 + 4.50T + 37T^{2} \) |
| 41 | \( 1 + 4.71T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 + 4.22T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 2.18T + 59T^{2} \) |
| 61 | \( 1 - 2.68T + 61T^{2} \) |
| 67 | \( 1 - 2.88T + 67T^{2} \) |
| 71 | \( 1 - 9.00T + 71T^{2} \) |
| 73 | \( 1 - 4.20T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 2.70T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
| 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
−10.01743453954818754656160105467, −9.004729405117878623452737724385, −8.455607929143334429825617037535, −7.913181486647085845023613437802, −6.78967458701960488267555069425, −5.76333451967706956360659303813, −4.99080316735972579070431826399, −3.61225763734995469468947103785, −1.91189006466009234453826698555, −1.06112232391071405361692778038,
1.06112232391071405361692778038, 1.91189006466009234453826698555, 3.61225763734995469468947103785, 4.99080316735972579070431826399, 5.76333451967706956360659303813, 6.78967458701960488267555069425, 7.913181486647085845023613437802, 8.455607929143334429825617037535, 9.004729405117878623452737724385, 10.01743453954818754656160105467
