L(s) = 1 | + (33.6 + 30.2i)2-s + (−145. − 145. i)3-s + (214. + 2.03e3i)4-s + (−4.65e3 + 4.65e3i)5-s + (−486. − 9.27e3i)6-s − 4.32e4i·7-s + (−5.44e4 + 7.49e4i)8-s − 1.35e5i·9-s + (−2.97e5 + 1.56e4i)10-s + (−6.85e5 + 6.85e5i)11-s + (2.64e5 − 3.26e5i)12-s + (−1.40e6 − 1.40e6i)13-s + (1.30e6 − 1.45e6i)14-s + 1.34e6·15-s + (−4.10e6 + 8.72e5i)16-s + 3.84e6·17-s + ⋯ |
L(s) = 1 | + (0.743 + 0.669i)2-s + (−0.344 − 0.344i)3-s + (0.104 + 0.994i)4-s + (−0.665 + 0.665i)5-s + (−0.0255 − 0.486i)6-s − 0.972i·7-s + (−0.587 + 0.809i)8-s − 0.762i·9-s + (−0.940 + 0.0493i)10-s + (−1.28 + 1.28i)11-s + (0.306 − 0.378i)12-s + (−1.04 − 1.04i)13-s + (0.650 − 0.722i)14-s + 0.459·15-s + (−0.978 + 0.208i)16-s + 0.656·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.566i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.0703914 - 0.226609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0703914 - 0.226609i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-33.6 - 30.2i)T \) |
good | 3 | \( 1 + (145. + 145. i)T + 1.77e5iT^{2} \) |
| 5 | \( 1 + (4.65e3 - 4.65e3i)T - 4.88e7iT^{2} \) |
| 7 | \( 1 + 4.32e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (6.85e5 - 6.85e5i)T - 2.85e11iT^{2} \) |
| 13 | \( 1 + (1.40e6 + 1.40e6i)T + 1.79e12iT^{2} \) |
| 17 | \( 1 - 3.84e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (7.81e6 + 7.81e6i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 - 1.99e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-1.18e8 - 1.18e8i)T + 1.22e16iT^{2} \) |
| 31 | \( 1 - 5.32e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (3.70e8 - 3.70e8i)T - 1.77e17iT^{2} \) |
| 41 | \( 1 - 8.83e7iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-1.81e8 + 1.81e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 + 1.92e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (-1.39e9 + 1.39e9i)T - 9.26e18iT^{2} \) |
| 59 | \( 1 + (2.09e9 - 2.09e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (-1.23e9 - 1.23e9i)T + 4.35e19iT^{2} \) |
| 67 | \( 1 + (9.73e9 + 9.73e9i)T + 1.22e20iT^{2} \) |
| 71 | \( 1 - 2.30e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 2.18e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 1.21e8T + 7.47e20T^{2} \) |
| 83 | \( 1 + (4.69e10 + 4.69e10i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 - 6.68e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 4.52e10T + 7.15e21T^{2} \) |
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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
−17.24310743411184464809911354314, −15.49592157649862714566166386161, −14.79509464387958969476543859452, −13.08361679003986409935147727489, −12.08506220011877270880267259037, −10.38547013057894201964154252024, −7.66396031003038033503671126897, −6.94839516997619059053348053392, −4.92131942693315712579191969088, −3.14285079519994646074491927167,
0.07886282812753708948790843771, 2.46549648243321155017067791248, 4.53502952500212921927537633659, 5.68142834532944676920640840084, 8.368886730519757743827145059689, 10.25262502401925425436777354401, 11.61798240605425545026035708221, 12.58133161376158491090633196019, 14.10599558606606280572981317443, 15.67268087326054631945010255511