| L(s) = 1 | + (−0.204 − 1.28i)2-s + (−1.35 + 1.08i)3-s + (0.281 − 0.0915i)4-s + (1.07 + 0.548i)5-s + (1.67 + 1.52i)6-s + (3.44 − 0.827i)7-s + (−1.36 − 2.67i)8-s + (0.660 − 2.92i)9-s + (0.487 − 1.50i)10-s + (−0.784 + 0.918i)11-s + (−0.282 + 0.428i)12-s + (−0.178 − 0.291i)13-s + (−1.77 − 4.27i)14-s + (−2.05 + 0.422i)15-s + (−2.68 + 1.95i)16-s + (4.10 − 0.322i)17-s + ⋯ |
| L(s) = 1 | + (−0.144 − 0.911i)2-s + (−0.781 + 0.624i)3-s + (0.140 − 0.0457i)4-s + (0.481 + 0.245i)5-s + (0.681 + 0.621i)6-s + (1.30 − 0.312i)7-s + (−0.481 − 0.944i)8-s + (0.220 − 0.975i)9-s + (0.154 − 0.474i)10-s + (−0.236 + 0.276i)11-s + (−0.0814 + 0.123i)12-s + (−0.0495 − 0.0808i)13-s + (−0.473 − 1.14i)14-s + (−0.529 + 0.109i)15-s + (−0.671 + 0.487i)16-s + (0.995 − 0.0783i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.911458 - 0.420404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.911458 - 0.420404i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.35 - 1.08i)T \) |
| 41 | \( 1 + (2.95 - 5.68i)T \) |
| good | 2 | \( 1 + (0.204 + 1.28i)T + (-1.90 + 0.618i)T^{2} \) |
| 5 | \( 1 + (-1.07 - 0.548i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (-3.44 + 0.827i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (0.784 - 0.918i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.178 + 0.291i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-4.10 + 0.322i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (3.81 - 6.21i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-1.27 - 0.923i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (5.66 + 0.445i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (3.68 + 1.19i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.78 - 5.49i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (7.41 - 1.17i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (0.345 - 1.43i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-1.03 + 13.1i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (3.03 - 4.17i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.74 + 11.0i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-8.31 - 9.73i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-3.32 - 2.84i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-7.90 + 7.90i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.514 + 1.24i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 2.44iT - 83T^{2} \) |
| 89 | \( 1 + (-2.38 - 9.91i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (-3.17 + 2.71i)T + (15.1 - 95.8i)T^{2} \) |
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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
−12.91007709778264522598669740257, −11.87358101590312748214239107386, −11.18580125140405886003432639829, −10.29196149634960393075744325476, −9.734474866768543804116058044296, −7.979352213784538869745536698241, −6.42962905480706231700183730567, −5.21760147769417007679110839017, −3.74360361801732104592597845761, −1.70974932383357855719031841923,
2.02675950701513873098333668566, 5.08499686212259552655400567010, 5.73710469792146142335796414255, 7.03770863018311331647571246453, 7.910606497980570831487129282865, 8.945935233155097642981252445241, 10.81971237518803573222991729385, 11.45143892327989319038328175927, 12.51461447885706005862073037388, 13.65869021521685721663474931680
