| L(s) = 1 | + (1.23 + 3.80i)2-s + (7.28 − 5.29i)3-s + (−12.9 + 9.40i)4-s + (25.5 + 49.7i)5-s + (29.1 + 21.1i)6-s + 186.·7-s + (−51.7 − 37.6i)8-s + (25.0 − 77.0i)9-s + (−157. + 158. i)10-s + (150. + 463. i)11-s + (−44.4 + 136. i)12-s + (307. − 945. i)13-s + (230. + 710. i)14-s + (449. + 226. i)15-s + (79.1 − 243. i)16-s + (−942. − 684. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.456 + 0.889i)5-s + (0.330 + 0.239i)6-s + 1.43·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.498 + 0.501i)10-s + (0.375 + 1.15i)11-s + (−0.0892 + 0.274i)12-s + (0.504 − 1.55i)13-s + (0.314 + 0.968i)14-s + (0.515 + 0.260i)15-s + (0.0772 − 0.237i)16-s + (−0.790 − 0.574i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.169828157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.169828157\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 - 3.80i)T \) |
| 3 | \( 1 + (-7.28 + 5.29i)T \) |
| 5 | \( 1 + (-25.5 - 49.7i)T \) |
| good | 7 | \( 1 - 186.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-150. - 463. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-307. + 945. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (942. + 684. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-917. - 666. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-512. - 1.57e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (3.08e3 - 2.24e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-8.36e3 - 6.07e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.20e3 + 3.71e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.70e3 + 5.23e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.39e4 - 1.01e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.77e4 - 1.29e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (5.77e3 - 1.77e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.08e4 - 3.32e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (4.48e4 + 3.26e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-4.32e4 + 3.13e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.88e4 + 5.80e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (3.48e4 - 2.53e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (8.38e4 + 6.09e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (2.01e4 + 6.18e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.49e4 + 1.08e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.51790693326606254029651220482, −11.32276346678692101314864459153, −10.24671511568630571962002254452, −9.010717574489179517395228620144, −7.79381532328681060385512351021, −7.22117740549711828778838908209, −5.86837339094758946618382768392, −4.66334168410943371414011338179, −3.03137153941831486893882344068, −1.53080412867938504231396631412,
1.11066755974718133942937937585, 2.18798110148784509690298636784, 4.07859624533884589039305804998, 4.81186879258582864337214150480, 6.19619862794451293167476636405, 8.245106309711476033953171534646, 8.785566192307172681839261738317, 9.748313477595109641521215604240, 11.33026209991159025662797789772, 11.45270616202002491273539033385
