L(s) = 1 | − 8.10·2-s − 4.25·3-s + 1.75·4-s + 55.9i·5-s + 34.4·6-s − 124. i·7-s + 504.·8-s − 710.·9-s − 453. i·10-s + 346. i·11-s − 7.46·12-s − 1.90e3·13-s + 1.01e3i·14-s − 237. i·15-s − 4.20e3·16-s + 6.96e3i·17-s + ⋯ |
L(s) = 1 | − 1.01·2-s − 0.157·3-s + 0.0274·4-s + 0.447i·5-s + 0.159·6-s − 0.363i·7-s + 0.985·8-s − 0.975·9-s − 0.453i·10-s + 0.260i·11-s − 0.00431·12-s − 0.866·13-s + 0.368i·14-s − 0.0704i·15-s − 1.02·16-s + 1.41i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.5009181088\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5009181088\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 55.9iT \) |
| 23 | \( 1 + (8.74e3 - 8.46e3i)T \) |
good | 2 | \( 1 + 8.10T + 64T^{2} \) |
| 3 | \( 1 + 4.25T + 729T^{2} \) |
| 7 | \( 1 + 124. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 346. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.90e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 6.96e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.80e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 1.91e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.31e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.83e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 7.75e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.01e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.93e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.77e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.81e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.17e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.55e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 2.00e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 5.55e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.97e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.47e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 4.86e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.66e6iT - 8.32e11T^{2} \) |
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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
−12.04923051431030940822587364798, −10.88465960829538875078766809461, −10.15606176846462159795582749551, −9.076496230629009795373282696896, −8.035491074547250081066301255237, −7.05419621409274456166969802043, −5.55541757775997264339463537290, −3.98031442207600380267828668440, −2.12977590733226866906011357408, −0.36667098400655784425841420163,
0.75025086285265408142388611098, 2.53972757940844834009788544055, 4.57202292199668815578999268412, 5.75670777133141631980524840274, 7.41218206310465197740655930255, 8.433886285953477908269285762508, 9.244295707096614554569903054535, 10.16983658173575829415159072673, 11.42514123418909467835675725630, 12.29229969962458599290967442165