| L(s) = 1 | + (−7.64 − 13.5i)3-s + (40.7 − 70.5i)5-s + (−89.6 − 155. i)7-s + (−126. + 207. i)9-s + (−250. − 433. i)11-s + (275. − 476. i)13-s + (−1.26e3 − 13.9i)15-s + 753.·17-s + 2.57e3·19-s + (−1.42e3 + 2.40e3i)21-s + (−1.37e3 + 2.37e3i)23-s + (−1.75e3 − 3.03e3i)25-s + (3.78e3 + 124. i)27-s + (−1.95e3 − 3.38e3i)29-s + (−1.55e3 + 2.68e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.490 − 0.871i)3-s + (0.728 − 1.26i)5-s + (−0.691 − 1.19i)7-s + (−0.518 + 0.854i)9-s + (−0.623 − 1.08i)11-s + (0.451 − 0.782i)13-s + (−1.45 − 0.0159i)15-s + 0.632·17-s + 1.63·19-s + (−0.704 + 1.18i)21-s + (−0.541 + 0.937i)23-s + (−0.561 − 0.972i)25-s + (0.999 + 0.0329i)27-s + (−0.431 − 0.747i)29-s + (−0.290 + 0.502i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.288663706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288663706\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (7.64 + 13.5i)T \) |
| good | 5 | \( 1 + (-40.7 + 70.5i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (89.6 + 155. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (250. + 433. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-275. + 476. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 753.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.37e3 - 2.37e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.95e3 + 3.38e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.55e3 - 2.68e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 9.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.11e3 - 1.92e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-7.14e3 - 1.23e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (3.23e3 + 5.60e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 1.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-2.85e3 + 4.94e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-5.89e3 - 1.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.77e3 - 3.06e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 5.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.78e4 + 4.81e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.99e4 - 3.46e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-8.29e4 - 1.43e5i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−11.78455685868298407467354573328, −10.58627804508454487112214582587, −9.624098154159341388672965476677, −8.261264773459213349776297646044, −7.37916087384058821069670458665, −5.88644431118048269362308796943, −5.29439144854471304479156698728, −3.34473583515559701086466866646, −1.27609543811733971513064205197, −0.50802988034308700843044780301,
2.31879247421945912971925094683, 3.45584533247781661691158638066, 5.24322182968681034420254313645, 6.09304603019625152635984065439, 7.13506179223991148402903896829, 9.036762412598472563044462125735, 9.833392216931160307748509075770, 10.49186257109720109216716495526, 11.68073927373701746211480198565, 12.50225296918609323377619987197
