L(s) = 1 | + 64i·2-s − 180. i·3-s − 4.09e3·4-s + (3.33e4 − 1.04e4i)5-s + 1.15e4·6-s + 3.86e5i·7-s − 2.62e5i·8-s + 1.56e6·9-s + (6.71e5 + 2.13e6i)10-s − 8.76e6·11-s + 7.38e5i·12-s + 3.09e7i·13-s − 2.47e7·14-s + (−1.89e6 − 6.00e6i)15-s + 1.67e7·16-s + 1.18e8i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.142i·3-s − 0.5·4-s + (0.953 − 0.300i)5-s + 0.100·6-s + 1.24i·7-s − 0.353i·8-s + 0.979·9-s + (0.212 + 0.674i)10-s − 1.49·11-s + 0.0714i·12-s + 1.77i·13-s − 0.878·14-s + (−0.0428 − 0.136i)15-s + 0.250·16-s + 1.19i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(1.04866 + 1.42966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04866 + 1.42966i\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 64iT \) |
| 5 | \( 1 + (-3.33e4 + 1.04e4i)T \) |
good | 3 | \( 1 + 180. iT - 1.59e6T^{2} \) |
| 7 | \( 1 - 3.86e5iT - 9.68e10T^{2} \) |
| 11 | \( 1 + 8.76e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 3.09e7iT - 3.02e14T^{2} \) |
| 17 | \( 1 - 1.18e8iT - 9.90e15T^{2} \) |
| 19 | \( 1 - 9.77e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 2.65e8iT - 5.04e17T^{2} \) |
| 29 | \( 1 - 3.94e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 4.11e8T + 2.44e19T^{2} \) |
| 37 | \( 1 - 4.34e9iT - 2.43e20T^{2} \) |
| 41 | \( 1 + 4.06e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.14e10iT - 1.71e21T^{2} \) |
| 47 | \( 1 + 1.15e11iT - 5.46e21T^{2} \) |
| 53 | \( 1 + 1.13e11iT - 2.60e22T^{2} \) |
| 59 | \( 1 + 7.68e10T + 1.04e23T^{2} \) |
| 61 | \( 1 - 1.14e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.23e12iT - 5.48e23T^{2} \) |
| 71 | \( 1 - 6.95e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 3.96e11iT - 1.67e24T^{2} \) |
| 79 | \( 1 - 5.70e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.89e12iT - 8.87e24T^{2} \) |
| 89 | \( 1 + 4.80e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 3.52e12iT - 6.73e25T^{2} \) |
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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
−18.10219427535810122991533351813, −16.47717248740048958398271475605, −15.30696340272417674857691913611, −13.66709847064020902810497777160, −12.44913938356477564050736242793, −9.971185789436909102501634881826, −8.537414012898196351336020805840, −6.48909561852359795711573555200, −4.98878442543831491806080866309, −1.96512125483128980072436398489,
0.862042731152666796046280471896, 2.93357209770896062760449975051, 5.08868911884824691715741164096, 7.51528786160312530224685357477, 10.01756725321176748574904647156, 10.55391418521005433996944968417, 12.94705682707130235333420559588, 13.76316255767835613062430001825, 15.74406592716377880677574955138, 17.59034529098098488892804616429