| L(s) = 1 | + (−0.992 − 0.810i)2-s + (0.0564 + 0.00455i)3-s + (−0.0714 − 0.349i)4-s + (0.992 − 0.120i)5-s + (−0.0523 − 0.0502i)6-s + (1.35 + 2.14i)7-s + (−1.40 + 2.67i)8-s + (−2.95 − 0.480i)9-s + (−1.08 − 0.685i)10-s + (−0.236 − 1.45i)11-s + (−0.00243 − 0.0200i)12-s + (3.11 + 1.82i)13-s + (0.392 − 3.23i)14-s + (0.0565 − 0.00227i)15-s + (2.90 − 1.23i)16-s + (−5.74 + 3.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.702 − 0.573i)2-s + (0.0325 + 0.00262i)3-s + (−0.0357 − 0.174i)4-s + (0.443 − 0.0539i)5-s + (−0.0213 − 0.0205i)6-s + (0.512 + 0.811i)7-s + (−0.496 + 0.945i)8-s + (−0.985 − 0.160i)9-s + (−0.342 − 0.216i)10-s + (−0.0711 − 0.437i)11-s + (−0.000703 − 0.00579i)12-s + (0.863 + 0.505i)13-s + (0.104 − 0.863i)14-s + (0.0146 − 0.000588i)15-s + (0.726 − 0.309i)16-s + (−1.39 + 0.881i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.846258 + 0.249667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.846258 + 0.249667i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.992 + 0.120i)T \) |
| 13 | \( 1 + (-3.11 - 1.82i)T \) |
| good | 2 | \( 1 + (0.992 + 0.810i)T + (0.400 + 1.95i)T^{2} \) |
| 3 | \( 1 + (-0.0564 - 0.00455i)T + (2.96 + 0.481i)T^{2} \) |
| 7 | \( 1 + (-1.35 - 2.14i)T + (-3.00 + 6.32i)T^{2} \) |
| 11 | \( 1 + (0.236 + 1.45i)T + (-10.4 + 3.48i)T^{2} \) |
| 17 | \( 1 + (5.74 - 3.63i)T + (7.28 - 15.3i)T^{2} \) |
| 19 | \( 1 + (-4.17 - 2.41i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.21 - 2.10i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.96 - 4.85i)T + (-5.80 - 28.4i)T^{2} \) |
| 31 | \( 1 + (1.69 - 6.88i)T + (-27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (5.60 - 1.62i)T + (31.2 - 19.7i)T^{2} \) |
| 41 | \( 1 + (-0.414 + 5.13i)T + (-40.4 - 6.57i)T^{2} \) |
| 43 | \( 1 + (-0.222 + 0.766i)T + (-36.3 - 22.9i)T^{2} \) |
| 47 | \( 1 + (-2.72 + 3.07i)T + (-5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (-9.20 - 4.83i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (2.25 - 5.29i)T + (-40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (0.103 - 2.56i)T + (-60.8 - 4.90i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 2.30i)T + (61.6 + 26.2i)T^{2} \) |
| 71 | \( 1 + (2.99 - 1.42i)T + (44.9 - 54.9i)T^{2} \) |
| 73 | \( 1 + (-11.5 - 4.36i)T + (54.6 + 48.4i)T^{2} \) |
| 79 | \( 1 + (0.208 + 0.184i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (-7.17 + 4.94i)T + (29.4 - 77.6i)T^{2} \) |
| 89 | \( 1 + (8.80 - 5.08i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.800 - 1.06i)T + (-26.9 - 93.1i)T^{2} \) |
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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
−10.41597882611789012774324083976, −9.082196167967179256048310110032, −8.904084201155309759388433806329, −8.273975338775447194055329295693, −6.73865527972751896507762203472, −5.64762240345252636726228788113, −5.33894680126546760090942350027, −3.59705955747789265925524988170, −2.35829829693663725092940915134, −1.42659018539802433653230384498,
0.56412441961616416899915131030, 2.44311536807878870578187162099, 3.68331173043329799331289426521, 4.84241664271820712260318616520, 5.95458236220644535726427412037, 6.89830818772069610962580624696, 7.62496395464881612994935844056, 8.385181242864122588372973768683, 9.168051250001994357779260258699, 9.825810345538520284437952792757
