L(s) = 1 | + (1.36 − 2.36i)2-s + (0.267 + 0.464i)4-s + (2.5 + 4.33i)5-s + (−3.46 + 6i)7-s + 23.3·8-s + 13.6·10-s + (−29.7 + 51.5i)11-s + (12.5 + 21.7i)13-s + (9.46 + 16.3i)14-s + (29.7 − 51.4i)16-s − 112.·17-s − 122.·19-s + (−1.33 + 2.32i)20-s + (81.2 + 140. i)22-s + (−48.7 − 84.3i)23-s + ⋯ |
L(s) = 1 | + (0.482 − 0.836i)2-s + (0.0334 + 0.0580i)4-s + (0.223 + 0.387i)5-s + (−0.187 + 0.323i)7-s + 1.03·8-s + 0.431·10-s + (−0.815 + 1.41i)11-s + (0.267 + 0.463i)13-s + (0.180 + 0.312i)14-s + (0.464 − 0.804i)16-s − 1.61·17-s − 1.48·19-s + (−0.0149 + 0.0259i)20-s + (0.787 + 1.36i)22-s + (−0.441 − 0.764i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.757153303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.757153303\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 2 | \( 1 + (-1.36 + 2.36i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (3.46 - 6i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (29.7 - 51.5i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-12.5 - 21.7i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (48.7 + 84.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (63.2 - 109. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-54.1 - 93.7i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (102. + 178. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-40.0 + 69.4i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (134. - 233. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 98.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + (152. + 263. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (333. - 577. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-317. - 550. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 751.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (11.9 - 20.6i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-408. + 707. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 513T + 7.04e5T^{2} \) |
| 97 | \( 1 + (214. - 370. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.89628752800759980851599179842, −10.55155443659779880208880913440, −9.421456220341571143974813700561, −8.324815459575570082035092075675, −7.16479821745746163518956722248, −6.38176949578361408522547059953, −4.79381176496572232938296037058, −4.09446338697855387477604490130, −2.53882282175257105440461742054, −2.02557247024823828750165116459,
0.45152989022130601893063418624, 2.20974606218263046685825037031, 3.89839590346193349424048504950, 4.95420070529149909830965651174, 5.99624135165016064622180536905, 6.49565896009283313404144246146, 7.86258558337101181891837649684, 8.467075429554838304959737176602, 9.734688770608373359030011878439, 10.81368783086150474873747887019