| L(s) = 1 | − 5.06i·3-s + 25i·5-s + 250.·7-s + 217.·9-s + 94.5i·11-s − 887. i·13-s + 126.·15-s + 1.44e3·17-s + 611. i·19-s − 1.26e3i·21-s − 814.·23-s − 625·25-s − 2.33e3i·27-s + 5.29e3i·29-s − 3.61e3·31-s + ⋯ |
| L(s) = 1 | − 0.325i·3-s + 0.447i·5-s + 1.93·7-s + 0.894·9-s + 0.235i·11-s − 1.45i·13-s + 0.145·15-s + 1.21·17-s + 0.388i·19-s − 0.627i·21-s − 0.320·23-s − 0.200·25-s − 0.615i·27-s + 1.16i·29-s − 0.675·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.064138908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.064138908\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25iT \) |
| good | 3 | \( 1 + 5.06iT - 243T^{2} \) |
| 7 | \( 1 - 250.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 94.5iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 887. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.44e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 611. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 814.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.29e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.48e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 9.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.02e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.30e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.66e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 715. iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.19e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.15e4T + 8.58e9T^{2} \) |
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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
−10.55953023232319546943601961745, −10.22981426875866608148929035398, −8.608447805088930817301581628614, −7.74641471957432144167207028493, −7.25350409031452855272428370103, −5.66865529597212396338259519877, −4.86242426517386049672523910272, −3.52467638617812078764266888528, −1.95650767505735235070314941674, −1.03989998118181387591201969539,
1.13378599577057286660540389328, 2.00120589253765951849134538930, 4.04411149565757320306835550340, 4.65886007948989309192326883115, 5.67367970867608810522913248046, 7.21629324772729362540321894304, 7.989157152651081982781317022840, 8.958318411962156401868275920165, 9.860160100984897686133384046549, 10.98404525750624751996822335748
