| L(s) = 1 | + (3.23 − 2.35i)2-s + (−2.78 + 8.55i)3-s + (4.94 − 15.2i)4-s + (22.2 − 51.2i)5-s + (11.1 + 34.2i)6-s + 175.·7-s + (−19.7 − 60.8i)8-s + (−65.5 − 47.6i)9-s + (−48.4 − 218. i)10-s + (−42.6 + 30.9i)11-s + (116. + 84.6i)12-s + (−226. − 164. i)13-s + (568. − 413. i)14-s + (376. + 333. i)15-s + (−207. − 150. i)16-s + (151. + 467. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (0.398 − 0.917i)5-s + (0.126 + 0.388i)6-s + 1.35·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.153 − 0.690i)10-s + (−0.106 + 0.0771i)11-s + (0.233 + 0.169i)12-s + (−0.371 − 0.270i)13-s + (0.775 − 0.563i)14-s + (0.432 + 0.382i)15-s + (−0.202 − 0.146i)16-s + (0.127 + 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.952782097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.952782097\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 3 | \( 1 + (2.78 - 8.55i)T \) |
| 5 | \( 1 + (-22.2 + 51.2i)T \) |
| good | 7 | \( 1 - 175.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (42.6 - 30.9i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (226. + 164. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-151. - 467. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (108. + 332. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-2.54e3 + 1.84e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-1.99e3 + 6.12e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.50e3 + 4.64e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-6.07e3 - 4.41e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (73.8 + 53.6i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.67e3 + 8.22e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-475. + 1.46e3i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-371. - 269. i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (3.75e4 - 2.73e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.61e4 + 4.96e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (6.47e3 - 1.99e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.48e4 - 1.07e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.51e4 - 7.74e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.39e4 - 4.28e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-4.88e4 + 3.55e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (3.12e4 - 9.60e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−11.87494442463774710174910415755, −11.01140741380643633679628077402, −10.01749390024539741562776594432, −8.912054682964995560666472471574, −7.83282650956498717313768628767, −5.97750318808534606117163614894, −4.95055792550065522732511939398, −4.29731786398749618734733608139, −2.35134132616051245964540766557, −0.901613988619168239948865707536,
1.61506065828270290607994547706, 2.99363425644914776817439777497, 4.76951652029097425485467020342, 5.77932677557577423822673065617, 7.04963508096146409437858653801, 7.68314549753639055726632352395, 9.033337690297750981817811109429, 10.67963656872153207427597059254, 11.35563972819235152995845287004, 12.36106584114437601757653849899
