L(s) = 1 | + 0.822·2-s − 0.874·3-s − 1.32·4-s + 0.00515·5-s − 0.718·6-s + 3.88·7-s − 2.73·8-s − 2.23·9-s + 0.00423·10-s + 1.15·12-s − 2.53·13-s + 3.19·14-s − 0.00450·15-s + 0.400·16-s − 2.29·17-s − 1.83·18-s + 0.555·19-s − 0.00682·20-s − 3.39·21-s − 4.57·23-s + 2.38·24-s − 4.99·25-s − 2.08·26-s + 4.57·27-s − 5.14·28-s − 1.66·29-s − 0.00370·30-s + ⋯ |
L(s) = 1 | + 0.581·2-s − 0.504·3-s − 0.661·4-s + 0.00230·5-s − 0.293·6-s + 1.46·7-s − 0.966·8-s − 0.745·9-s + 0.00134·10-s + 0.334·12-s − 0.703·13-s + 0.853·14-s − 0.00116·15-s + 0.100·16-s − 0.556·17-s − 0.433·18-s + 0.127·19-s − 0.00152·20-s − 0.741·21-s − 0.954·23-s + 0.487·24-s − 0.999·25-s − 0.408·26-s + 0.880·27-s − 0.972·28-s − 0.309·29-s − 0.000676·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.310186366\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.310186366\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 - 0.822T + 2T^{2} \) |
| 3 | \( 1 + 0.874T + 3T^{2} \) |
| 5 | \( 1 - 0.00515T + 5T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 + 2.29T + 17T^{2} \) |
| 19 | \( 1 - 0.555T + 19T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 + 6.23T + 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 + 0.618T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 - 6.86T + 47T^{2} \) |
| 53 | \( 1 + 7.04T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.350T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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
−7.951209892403933064688172344717, −7.29748156987970325966577050004, −6.10192446798509215050280403773, −5.71023892292735030699187856118, −5.05799301860348794336120997201, −4.44719438193902315915589713181, −3.87192688766877680770068184768, −2.71097215836843967297788452626, −1.88791882728173660330044232495, −0.53125349456294764969792744325,
0.53125349456294764969792744325, 1.88791882728173660330044232495, 2.71097215836843967297788452626, 3.87192688766877680770068184768, 4.44719438193902315915589713181, 5.05799301860348794336120997201, 5.71023892292735030699187856118, 6.10192446798509215050280403773, 7.29748156987970325966577050004, 7.951209892403933064688172344717