| L(s) = 1 | + (0.695 − 1.23i)2-s + (0.0311 − 0.0752i)3-s + (−1.03 − 1.71i)4-s + (−0.453 + 0.891i)5-s + (−0.0710 − 0.0907i)6-s + (0.547 + 2.28i)7-s + (−2.82 + 0.0818i)8-s + (2.11 + 2.11i)9-s + (0.781 + 1.17i)10-s + (1.38 − 1.61i)11-s + (−0.161 + 0.0243i)12-s + (1.49 + 2.43i)13-s + (3.19 + 0.912i)14-s + (0.0529 + 0.0619i)15-s + (−1.86 + 3.53i)16-s + (0.245 + 3.11i)17-s + ⋯ |
| L(s) = 1 | + (0.491 − 0.870i)2-s + (0.0180 − 0.0434i)3-s + (−0.516 − 0.856i)4-s + (−0.203 + 0.398i)5-s + (−0.0289 − 0.0370i)6-s + (0.207 + 0.862i)7-s + (−0.999 + 0.0289i)8-s + (0.705 + 0.705i)9-s + (0.247 + 0.372i)10-s + (0.416 − 0.487i)11-s + (−0.0465 + 0.00703i)12-s + (0.414 + 0.676i)13-s + (0.853 + 0.243i)14-s + (0.0136 + 0.0159i)15-s + (−0.466 + 0.884i)16-s + (0.0595 + 0.756i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.91804 - 0.278227i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91804 - 0.278227i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.695 + 1.23i)T \) |
| 5 | \( 1 + (0.453 - 0.891i)T \) |
| 41 | \( 1 + (-5.89 - 2.49i)T \) |
| good | 3 | \( 1 + (-0.0311 + 0.0752i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.547 - 2.28i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-1.38 + 1.61i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.49 - 2.43i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.245 - 3.11i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-0.360 - 0.221i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-2.47 - 1.79i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.147 - 1.87i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-2.51 + 7.74i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.301 + 0.928i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.843 - 5.32i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.67 + 6.97i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (1.78 + 0.140i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (7.56 - 10.4i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.42 + 8.98i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (2.58 - 2.20i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (9.58 + 8.18i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-3.83 + 3.83i)T - 73iT^{2} \) |
| 79 | \( 1 + (1.07 + 0.445i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 4.65iT - 83T^{2} \) |
| 89 | \( 1 + (10.1 - 2.44i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-7.07 + 6.04i)T + (15.1 - 95.8i)T^{2} \) |
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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
−10.41441823082273901884643584817, −9.450819071059292757934210059255, −8.734336847724446772917932164835, −7.71764841772847034737376807875, −6.44927952237287552966330096907, −5.70663185330137650764884307457, −4.59579551890121026755497140930, −3.74202629553648723611767254490, −2.54430988883015638441938329829, −1.49702414492854335162067817697,
0.947946559011202808498460728221, 3.15600787818696600450358030887, 4.17124666585823166392852681171, 4.76844371463089367328427619916, 5.93171382731033705396193817171, 6.99446056852254617539374244080, 7.40178509602953047775657960058, 8.464436658700319775364377489077, 9.253229226732444719675648862161, 10.11311463770056987656201461582
