| L(s) = 1 | + (−1.11 − 0.867i)2-s + (0.668 − 1.61i)3-s + (0.495 + 1.93i)4-s + (−0.453 + 0.891i)5-s + (−2.14 + 1.22i)6-s + (−0.195 − 0.812i)7-s + (1.12 − 2.59i)8-s + (−0.0381 − 0.0381i)9-s + (1.27 − 0.601i)10-s + (−1.75 + 2.05i)11-s + (3.45 + 0.496i)12-s + (2.32 + 3.79i)13-s + (−0.486 + 1.07i)14-s + (1.13 + 1.32i)15-s + (−3.50 + 1.91i)16-s + (0.465 + 5.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.789 − 0.613i)2-s + (0.386 − 0.932i)3-s + (0.247 + 0.968i)4-s + (−0.203 + 0.398i)5-s + (−0.876 + 0.499i)6-s + (−0.0737 − 0.307i)7-s + (0.398 − 0.917i)8-s + (−0.0127 − 0.0127i)9-s + (0.404 − 0.190i)10-s + (−0.528 + 0.618i)11-s + (0.998 + 0.143i)12-s + (0.644 + 1.05i)13-s + (−0.130 + 0.287i)14-s + (0.293 + 0.343i)15-s + (−0.877 + 0.479i)16-s + (0.112 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04094 + 0.0210304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04094 + 0.0210304i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.11 + 0.867i)T \) |
| 5 | \( 1 + (0.453 - 0.891i)T \) |
| 41 | \( 1 + (6.24 - 1.40i)T \) |
| good | 3 | \( 1 + (-0.668 + 1.61i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (0.195 + 0.812i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (1.75 - 2.05i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.32 - 3.79i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.465 - 5.91i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (1.24 + 0.761i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-0.253 - 0.184i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.0107 - 0.136i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (0.671 - 2.06i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.80 - 5.56i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-1.58 - 10.0i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.145 + 0.604i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (1.09 + 0.0858i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (-2.18 + 3.00i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.325 - 2.05i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-11.4 + 9.77i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (1.67 + 1.43i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-6.84 + 6.84i)T - 73iT^{2} \) |
| 79 | \( 1 + (-7.18 - 2.97i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 1.38iT - 83T^{2} \) |
| 89 | \( 1 + (10.5 - 2.52i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-11.8 + 10.1i)T + (15.1 - 95.8i)T^{2} \) |
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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
−10.33464055626659627881723718705, −9.411056849838736608828042623382, −8.323456820480518613431176821321, −7.930247679557527862940393523033, −6.92308583405685197572738371818, −6.45268257754484795403097777034, −4.52522496152669218408358455732, −3.50043637911458761006571183632, −2.26037800450074516636944656828, −1.42120747096866199325862917524,
0.68527794608567924418055106682, 2.64748098658045591506439078296, 3.85639465378870488667296397319, 5.13658254149195420382206014254, 5.68274247936315574701592404732, 6.94782398735441443822099137584, 7.920024003817281062181121866816, 8.668118913102118162591062113352, 9.227114992102050428264964800002, 10.07314743163206182663085397220
