| L(s) = 1 | + (1.61 + 1.17i)2-s + 3.80i·3-s + (1.23 + 3.80i)4-s + (−4.47 + 6.15i)6-s − 8.50i·7-s + (−2.47 + 7.60i)8-s − 5.47·9-s − 1.79i·11-s + (−14.4 + 4.70i)12-s − 0.472·13-s + (10 − 13.7i)14-s + (−12.9 + 9.40i)16-s + 23.8·17-s + (−8.85 − 6.43i)18-s + 9.40i·19-s + ⋯ |
| L(s) = 1 | + (0.809 + 0.587i)2-s + 1.26i·3-s + (0.309 + 0.951i)4-s + (−0.745 + 1.02i)6-s − 1.21i·7-s + (−0.309 + 0.951i)8-s − 0.608·9-s − 0.163i·11-s + (−1.20 + 0.391i)12-s − 0.0363·13-s + (0.714 − 0.983i)14-s + (−0.809 + 0.587i)16-s + 1.40·17-s + (−0.491 − 0.357i)18-s + 0.494i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.17905 + 1.62282i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17905 + 1.62282i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 - 1.17i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.80iT - 9T^{2} \) |
| 7 | \( 1 + 8.50iT - 49T^{2} \) |
| 11 | \( 1 + 1.79iT - 121T^{2} \) |
| 13 | \( 1 + 0.472T + 169T^{2} \) |
| 17 | \( 1 - 23.8T + 289T^{2} \) |
| 19 | \( 1 - 9.40iT - 361T^{2} \) |
| 23 | \( 1 + 16.1iT - 529T^{2} \) |
| 29 | \( 1 - 6.94T + 841T^{2} \) |
| 31 | \( 1 + 47.4iT - 961T^{2} \) |
| 37 | \( 1 + 26.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 2.00iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 21.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 88.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 21.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 39.1T + 9.40e3T^{2} \) |
| 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
−14.19941172920596749574290960532, −13.17294487179482291249324440804, −11.88992809024742380935981241084, −10.66602016039426855265629794201, −9.840792325751457165009211241472, −8.277074469338828202103589679353, −7.09735032931616127386747826722, −5.56902691100543768671113841495, −4.34637239266616286210119416659, −3.47487122353509333794313508761,
1.57979620373386552933294703489, 3.02321006338953953269158309629, 5.17846104479168738256011182350, 6.24473153711932484281349440134, 7.44935053307274165150480407729, 8.958853800838484393979751367048, 10.32342352103878287148475599599, 11.91438806903956871740257913345, 12.14435590734574919506438837763, 13.14440317037188030520461577881
