| L(s) = 1 | + (3.23 − 2.35i)2-s + (−2.78 + 8.55i)3-s + (4.94 − 15.2i)4-s + (−41.6 − 37.2i)5-s + (11.1 + 34.2i)6-s − 58.1·7-s + (−19.7 − 60.8i)8-s + (−65.5 − 47.6i)9-s + (−222. − 22.7i)10-s + (−75.5 + 54.8i)11-s + (116. + 84.6i)12-s + (756. + 549. i)13-s + (−188. + 136. i)14-s + (435. − 252. i)15-s + (−207. − 150. i)16-s + (665. + 2.04e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.744 − 0.667i)5-s + (0.126 + 0.388i)6-s − 0.448·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.703 − 0.0720i)10-s + (−0.188 + 0.136i)11-s + (0.233 + 0.169i)12-s + (1.24 + 0.902i)13-s + (−0.256 + 0.186i)14-s + (0.499 − 0.289i)15-s + (−0.202 − 0.146i)16-s + (0.558 + 1.71i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.478532098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.478532098\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (41.6 + 37.2i)T \) |
| good | 7 | \( 1 + 58.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + (75.5 - 54.8i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-756. - 549. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-665. - 2.04e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (315. + 971. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (270. - 196. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.97e3 - 6.07e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.52e3 - 4.67e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (4.10e3 + 2.98e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (2.94e3 + 2.13e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.27e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.80e3 - 2.09e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-3.43e3 + 1.05e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.92e4 - 2.85e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.41e4 + 1.02e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (3.54e3 + 1.09e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.67e4 + 5.15e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.37e4 - 1.72e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (3.99e3 - 1.22e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.26e4 - 6.98e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (8.31e4 - 6.04e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-2.63e4 + 8.10e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−12.39435370792037923740895180921, −11.26860256588042224379466724523, −10.56994279064069034270903724851, −9.240628193692361033941600578287, −8.357777702975526247756675029737, −6.69704709657472140042172513015, −5.48579517322740018726007229760, −4.22893406953770593884308871191, −3.46353006024840155107396537880, −1.36678500188582608644486146549,
0.45119175153800736858603806171, 2.76818604302373928548315682876, 3.83220567100205386887617803812, 5.52171641964657485941996978865, 6.50639762853564319869629594797, 7.54392468508997698752695135982, 8.323194079647795298576591037977, 10.01604522260802668267455345538, 11.26745734594988559144634161991, 11.92132925373476506176396037330
