L(s) = 1 | + (1.00 − 1.74i)2-s + (0.866 + 0.5i)3-s + (−1.02 − 1.77i)4-s + (1.28 + 1.83i)5-s + (1.74 − 1.00i)6-s + (−1.11 − 1.92i)7-s − 0.0905·8-s + (0.499 + 0.866i)9-s + (4.47 − 0.393i)10-s + (−4.40 − 2.54i)11-s − 2.04i·12-s + (1.22 + 3.38i)13-s − 4.47·14-s + (0.195 + 2.22i)15-s + (1.95 − 3.38i)16-s + (2.00 − 1.15i)17-s + ⋯ |
L(s) = 1 | + (0.711 − 1.23i)2-s + (0.499 + 0.288i)3-s + (−0.511 − 0.885i)4-s + (0.573 + 0.818i)5-s + (0.711 − 0.410i)6-s + (−0.420 − 0.727i)7-s − 0.0320·8-s + (0.166 + 0.288i)9-s + (1.41 − 0.124i)10-s + (−1.32 − 0.767i)11-s − 0.590i·12-s + (0.340 + 0.940i)13-s − 1.19·14-s + (0.0505 + 0.575i)15-s + (0.488 − 0.846i)16-s + (0.485 − 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64690 - 1.03566i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64690 - 1.03566i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-1.28 - 1.83i)T \) |
| 13 | \( 1 + (-1.22 - 3.38i)T \) |
good | 2 | \( 1 + (-1.00 + 1.74i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.11 + 1.92i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.40 + 2.54i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.00 + 1.15i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.07 - 3.50i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.77 + 2.75i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.503 + 0.872i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.55iT - 31T^{2} \) |
| 37 | \( 1 + (-0.579 + 1.00i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.42 - 0.823i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.97 + 4.60i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.34T + 47T^{2} \) |
| 53 | \( 1 + 9.36iT - 53T^{2} \) |
| 59 | \( 1 + (-0.894 + 0.516i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.32 - 9.22i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.66 - 2.88i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.54 + 0.889i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 4.49T + 73T^{2} \) |
| 79 | \( 1 + 7.05T + 79T^{2} \) |
| 83 | \( 1 - 2.25T + 83T^{2} \) |
| 89 | \( 1 + (7.03 + 4.06i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.01 + 5.22i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.46691463350141429933846873352, −11.16074412264008458474429270183, −10.39453058235763708620759883823, −10.05433427007758899858959586077, −8.477942254479492854471814253182, −7.13896928931004458285778552331, −5.75747941840560884043438163053, −4.21883412247453692251310267204, −3.24210336548136276797341692246, −2.14971155147749082983279931055,
2.38426236647866121480191841876, 4.34668689514578456511863571004, 5.54246352174206886684978120950, 6.15757373619547678983279772419, 7.64991379257234481080535490088, 8.269159481441469459614498203137, 9.453459218809863654171890854974, 10.53638999092796280647709252888, 12.47197602409513446121306426843, 12.95045914483003554096789770739