| L(s) = 1 | + (1.23 − 3.80i)2-s + (7.28 + 5.29i)3-s + (−12.9 − 9.40i)4-s + (−30.6 + 46.7i)5-s + (29.1 − 21.1i)6-s − 78.0·7-s + (−51.7 + 37.6i)8-s + (25.0 + 77.0i)9-s + (140. + 174. i)10-s + (120. − 372. i)11-s + (−44.4 − 136. i)12-s + (−103. − 317. i)13-s + (−96.4 + 296. i)14-s + (−470. + 178. i)15-s + (79.1 + 243. i)16-s + (705. − 512. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.547 + 0.836i)5-s + (0.330 − 0.239i)6-s − 0.601·7-s + (−0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + (0.442 + 0.551i)10-s + (0.301 − 0.927i)11-s + (−0.0892 − 0.274i)12-s + (−0.169 − 0.520i)13-s + (−0.131 + 0.404i)14-s + (−0.539 + 0.204i)15-s + (0.0772 + 0.237i)16-s + (0.591 − 0.430i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0341 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0341 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.804202910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804202910\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (30.6 - 46.7i)T \) |
| good | 7 | \( 1 + 78.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-120. + 372. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (103. + 317. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-705. + 512. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-2.25e3 + 1.63e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-817. + 2.51e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.80e3 - 3.48e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-593. + 431. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (679. + 2.09e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-735. - 2.26e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 5.24e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-5.60e3 - 4.07e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.74e4 + 1.99e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.55e4 + 4.78e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (3.76e3 - 1.15e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.56e3 - 2.58e3i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (4.70e4 + 3.41e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (7.40e3 - 2.27e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.34e4 - 9.77e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-5.10e4 + 3.71e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-9.49e3 + 2.92e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-5.48e4 - 3.98e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−11.72145789402286738529862589220, −10.87442886380762128594792163391, −9.967977322855529213883797524816, −8.960652488272526577323637308105, −7.71840827069414354048972825654, −6.44240000100238168846138176853, −4.90219551849788796616021624729, −3.33369855075674739422571338108, −2.90891281934789074692749456902, −0.60570555194686502525867113861,
1.29718283128688073124107692356, 3.38321359725365765153406687689, 4.53804279468866619866892881304, 5.89150752088352351170101568193, 7.24688955783852851943996941109, 7.934072657515801035016432888192, 9.159280384806896344024393293286, 9.873294615816037914570269397837, 11.92110973480577336301509606586, 12.35381215977652003866253595711
