L(s) = 1 | − 5.39i·2-s + 2.81·3-s − 21.1·4-s − 6.77i·5-s − 15.1i·6-s + 16.2·7-s + 70.7i·8-s − 19.0·9-s − 36.5·10-s + 56.6·11-s − 59.4·12-s − 41.6i·13-s − 87.7i·14-s − 19.0i·15-s + 212.·16-s + 68.5i·17-s + ⋯ |
L(s) = 1 | − 1.90i·2-s + 0.541·3-s − 2.63·4-s − 0.606i·5-s − 1.03i·6-s + 0.877·7-s + 3.12i·8-s − 0.706·9-s − 1.15·10-s + 1.55·11-s − 1.43·12-s − 0.888i·13-s − 1.67i·14-s − 0.328i·15-s + 3.32·16-s + 0.977i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.270609 - 1.29889i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.270609 - 1.29889i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (206. - 89.8i)T \) |
good | 2 | \( 1 + 5.39iT - 8T^{2} \) |
| 3 | \( 1 - 2.81T + 27T^{2} \) |
| 5 | \( 1 + 6.77iT - 125T^{2} \) |
| 7 | \( 1 - 16.2T + 343T^{2} \) |
| 11 | \( 1 - 56.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 68.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 84.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 70.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 111. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 322. iT - 2.97e4T^{2} \) |
| 41 | \( 1 - 131.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 112. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 98.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 215.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 40.1iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 720. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 476.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 664.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 281.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 706. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 818. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 63.4iT - 9.12e5T^{2} \) |
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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
−14.74427829605350781919074018896, −13.91990212907247305978799102254, −12.68191394123175853599222571520, −11.69057350094681813747364378603, −10.67738040920029770345049469035, −9.065632784024298439309767627271, −8.494506591339093111099969573045, −4.98234314011458615500957577835, −3.39205103370227674598536145932, −1.42947823421608526138806451791,
4.22204746749457819931203692813, 6.02807938425407256996016691887, 7.25307284704792828827910614884, 8.478552240628202469553752035736, 9.437984373326311268628279652721, 11.66866378929365963643004046299, 13.77621698555938224951671436187, 14.48199110544285619435665847305, 14.83746078056025791132702893919, 16.46666734377506745475136447377