| L(s) = 1 | + (−1.38 − 7.87i)2-s + (−2.39 + 13.6i)3-s + (−60.1 + 21.8i)4-s + (−34.7 − 29.1i)5-s + 110.·6-s + (−140. − 117. i)7-s + (256 + 443. i)8-s + (1.87e3 + 682. i)9-s + (−181. + 313. i)10-s + (−242. − 420. i)11-s + (−153. − 871. i)12-s + (381. − 138. i)13-s + (−731. + 1.26e3i)14-s + (479. − 402. i)15-s + (3.13e3 − 2.63e3i)16-s + (−1.25e4 − 4.55e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.0513 + 0.291i)3-s + (−0.469 + 0.171i)4-s + (−0.124 − 0.104i)5-s + 0.208·6-s + (−0.154 − 0.129i)7-s + (0.176 + 0.306i)8-s + (0.857 + 0.312i)9-s + (−0.0573 + 0.0992i)10-s + (−0.0550 − 0.0952i)11-s + (−0.0256 − 0.145i)12-s + (0.0481 − 0.0175i)13-s + (−0.0712 + 0.123i)14-s + (0.0366 − 0.0307i)15-s + (0.191 − 0.160i)16-s + (−0.618 − 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.368106 - 0.980570i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.368106 - 0.980570i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 7.87i)T \) |
| 37 | \( 1 + (-1.09e5 + 2.87e5i)T \) |
| good | 3 | \( 1 + (2.39 - 13.6i)T + (-2.05e3 - 747. i)T^{2} \) |
| 5 | \( 1 + (34.7 + 29.1i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (140. + 117. i)T + (1.43e5 + 8.11e5i)T^{2} \) |
| 11 | \( 1 + (242. + 420. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-381. + 138. i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.25e4 + 4.55e3i)T + (3.14e8 + 2.63e8i)T^{2} \) |
| 19 | \( 1 + (-1.37e3 + 7.77e3i)T + (-8.39e8 - 3.05e8i)T^{2} \) |
| 23 | \( 1 + (-3.45e4 + 5.98e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (6.75e4 + 1.16e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.68e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + (5.24e5 - 1.90e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + 6.32e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.61e5 + 2.79e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (3.91e5 - 3.28e5i)T + (2.03e11 - 1.15e12i)T^{2} \) |
| 59 | \( 1 + (-1.70e6 + 1.43e6i)T + (4.32e11 - 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-2.30e6 + 8.37e5i)T + (2.40e12 - 2.02e12i)T^{2} \) |
| 67 | \( 1 + (2.51e5 + 2.11e5i)T + (1.05e12 + 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-5.65e4 + 3.20e5i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 - 3.75e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (1.50e6 + 1.26e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-2.60e6 - 9.49e5i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (7.70e5 - 6.46e5i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-1.60e5 + 2.77e5i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.72952285834580002163348723596, −11.46247071526425522713487115158, −10.48170077712483905080184031039, −9.529844607596495701129907814752, −8.282991736847873881095269110577, −6.84029350173121728268244092385, −4.97484949696440542797581802981, −3.82594147084151037010968481648, −2.12988005010821298259916134381, −0.39208961465832784933795833784,
1.46545991108695645025032773951, 3.67816959488639912933972119993, 5.24975986405262590452420244370, 6.67251835063226838652714819105, 7.51515728537451161777018908320, 8.914879825573194583781761682403, 9.972384105906355909087997401694, 11.36552626120987448810610746394, 12.74252933019941766372271666424, 13.46761003142886480791480154949
