| L(s) = 1 | + (−5.16 − 6.11i)2-s + (23.0 + 14.0i)3-s + (−10.7 + 63.0i)4-s + (−44.3 + 25.5i)5-s + (−32.9 − 213. i)6-s + (23.6 + 13.6i)7-s + (441. − 260. i)8-s + (333. + 648. i)9-s + (385. + 138. i)10-s + (53.8 − 93.1i)11-s + (−1.13e3 + 1.30e3i)12-s + (−1.32e3 + 765. i)13-s + (−38.5 − 214. i)14-s + (−1.38e3 − 33.4i)15-s + (−3.86e3 − 1.35e3i)16-s − 2.17e3·17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 0.764i)2-s + (0.853 + 0.520i)3-s + (−0.167 + 0.985i)4-s + (−0.354 + 0.204i)5-s + (−0.152 − 0.988i)6-s + (0.0688 + 0.0397i)7-s + (0.861 − 0.507i)8-s + (0.457 + 0.889i)9-s + (0.385 + 0.138i)10-s + (0.0404 − 0.0700i)11-s + (−0.656 + 0.754i)12-s + (−0.603 + 0.348i)13-s + (−0.0140 − 0.0782i)14-s + (−0.409 − 0.00992i)15-s + (−0.943 − 0.330i)16-s − 0.442·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.546585 + 0.748870i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.546585 + 0.748870i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.16 + 6.11i)T \) |
| 3 | \( 1 + (-23.0 - 14.0i)T \) |
| good | 5 | \( 1 + (44.3 - 25.5i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-23.6 - 13.6i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-53.8 + 93.1i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.32e3 - 765. i)T + (2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 2.17e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 6.04e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (5.65e3 - 3.26e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.28e4 - 1.32e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (3.14e4 - 1.81e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 2.95e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-7.64e3 - 1.32e4i)T + (-2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (5.64e3 - 9.78e3i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-4.97e4 - 2.87e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 2.05e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (2.97e4 + 5.15e4i)T + (-2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (5.24e4 + 3.02e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.65e5 + 2.86e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.95e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 5.48e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-7.42e5 - 4.28e5i)T + (1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.05e5 + 5.28e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 5.29e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (1.94e5 - 3.36e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−13.67204011916019640559950878400, −12.53764214861330099825345834379, −11.28090387067202668718519389497, −10.29652177688876333853477655892, −9.240101504832764725381172460890, −8.291714927790265925121552100867, −7.15286092533104185505955117653, −4.55598906243762841296992245964, −3.31927168967840635954970320011, −1.94040224210843905380303975859,
0.38232667056630324900303082687, 2.15553064090307424991463623385, 4.34726705396745804087986623313, 6.20698936600269199838745039249, 7.44350702476268482120393344163, 8.290812053157177049474599406255, 9.295334258800632513296293124050, 10.49730191652340049541068183296, 12.13462541531894986243149449402, 13.34159497162034333294265883094
