| L(s) = 1 | + (−43.0 + 13.9i)2-s + (296. + 215. i)3-s + (1.65e3 − 1.20e3i)4-s + (8.35e3 − 2.57e4i)5-s + (−1.57e4 − 5.13e3i)6-s + (−2.52e4 − 3.47e4i)7-s + (−5.44e4 + 7.49e4i)8-s + (−1.22e5 − 3.77e5i)9-s + 1.22e6i·10-s + (−5.79e5 + 1.67e6i)11-s + 7.51e5·12-s + (5.13e5 − 1.66e5i)13-s + (1.57e6 + 1.14e6i)14-s + (8.02e6 − 5.83e6i)15-s + (1.29e6 − 3.98e6i)16-s + (−1.06e7 − 3.46e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.407 + 0.295i)3-s + (0.404 − 0.293i)4-s + (0.534 − 1.64i)5-s + (−0.338 − 0.109i)6-s + (−0.214 − 0.295i)7-s + (−0.207 + 0.286i)8-s + (−0.230 − 0.710i)9-s + 1.22i·10-s + (−0.327 + 0.944i)11-s + 0.251·12-s + (0.106 − 0.0345i)13-s + (0.208 + 0.151i)14-s + (0.704 − 0.512i)15-s + (0.0772 − 0.237i)16-s + (−0.442 − 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.251034 - 0.768998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.251034 - 0.768998i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (43.0 - 13.9i)T \) |
| 11 | \( 1 + (5.79e5 - 1.67e6i)T \) |
| good | 3 | \( 1 + (-296. - 215. i)T + (1.64e5 + 5.05e5i)T^{2} \) |
| 5 | \( 1 + (-8.35e3 + 2.57e4i)T + (-1.97e8 - 1.43e8i)T^{2} \) |
| 7 | \( 1 + (2.52e4 + 3.47e4i)T + (-4.27e9 + 1.31e10i)T^{2} \) |
| 13 | \( 1 + (-5.13e5 + 1.66e5i)T + (1.88e13 - 1.36e13i)T^{2} \) |
| 17 | \( 1 + (1.06e7 + 3.46e6i)T + (4.71e14 + 3.42e14i)T^{2} \) |
| 19 | \( 1 + (3.35e7 - 4.61e7i)T + (-6.83e14 - 2.10e15i)T^{2} \) |
| 23 | \( 1 + 1.15e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (5.32e8 + 7.33e8i)T + (-1.09e17 + 3.36e17i)T^{2} \) |
| 31 | \( 1 + (-4.73e7 - 1.45e8i)T + (-6.37e17 + 4.62e17i)T^{2} \) |
| 37 | \( 1 + (-2.13e9 + 1.55e9i)T + (2.03e18 - 6.26e18i)T^{2} \) |
| 41 | \( 1 + (1.96e9 - 2.70e9i)T + (-6.97e18 - 2.14e19i)T^{2} \) |
| 43 | \( 1 - 3.28e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-5.64e9 - 4.10e9i)T + (3.59e19 + 1.10e20i)T^{2} \) |
| 53 | \( 1 + (9.78e9 + 3.01e10i)T + (-3.97e20 + 2.88e20i)T^{2} \) |
| 59 | \( 1 + (4.80e10 - 3.48e10i)T + (5.49e20 - 1.69e21i)T^{2} \) |
| 61 | \( 1 + (-3.49e10 - 1.13e10i)T + (2.14e21 + 1.56e21i)T^{2} \) |
| 67 | \( 1 + 1.46e11T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-6.00e10 + 1.84e11i)T + (-1.32e22 - 9.64e21i)T^{2} \) |
| 73 | \( 1 + (3.40e10 + 4.68e10i)T + (-7.07e21 + 2.17e22i)T^{2} \) |
| 79 | \( 1 + (3.22e11 - 1.04e11i)T + (4.78e22 - 3.47e22i)T^{2} \) |
| 83 | \( 1 + (-7.18e10 - 2.33e10i)T + (8.64e22 + 6.28e22i)T^{2} \) |
| 89 | \( 1 - 6.46e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-4.31e11 - 1.32e12i)T + (-5.61e23 + 4.07e23i)T^{2} \) |
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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
−14.87506973336209669202074081819, −13.26134642495403325721127789226, −12.10780140480824980530096140863, −9.992660733334584421761013891557, −9.190747322211950402038886721442, −8.029312550821049401321671525133, −6.00657976831686418842905286797, −4.31271468356864544670733489299, −1.87077108139337935114144751838, −0.31937944388891995245608200814,
2.13694399037144127741965255697, 3.06120496775943806148073082170, 6.09611777322645046776199906198, 7.39010644354115328762151952677, 8.847986879816250884839703464694, 10.47952026178811369184814301912, 11.19547799911218986724384242749, 13.25911329765158068956591467034, 14.30590891234280563920573617123, 15.61361461026954024693995821503
