L(s) = 1 | + (−8.50 + 8.50i)2-s − 17.1·3-s − 80.5i·4-s + (73.6 − 73.6i)5-s + (145. − 145. i)6-s + (−214. − 214. i)7-s + (140. + 140. i)8-s − 436.·9-s + 1.25e3i·10-s + (231. + 231. i)11-s + 1.37e3i·12-s + (−2.07e3 − 708. i)13-s + 3.64e3·14-s + (−1.26e3 + 1.26e3i)15-s + 2.76e3·16-s − 7.52e3i·17-s + ⋯ |
L(s) = 1 | + (−1.06 + 1.06i)2-s − 0.633·3-s − 1.25i·4-s + (0.589 − 0.589i)5-s + (0.673 − 0.673i)6-s + (−0.624 − 0.624i)7-s + (0.275 + 0.275i)8-s − 0.598·9-s + 1.25i·10-s + (0.174 + 0.174i)11-s + 0.797i·12-s + (−0.946 − 0.322i)13-s + 1.32·14-s + (−0.373 + 0.373i)15-s + 0.673·16-s − 1.53i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0343 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0343 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.154823 - 0.160231i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.154823 - 0.160231i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (2.07e3 + 708. i)T \) |
good | 2 | \( 1 + (8.50 - 8.50i)T - 64iT^{2} \) |
| 3 | \( 1 + 17.1T + 729T^{2} \) |
| 5 | \( 1 + (-73.6 + 73.6i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + (214. + 214. i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 + (-231. - 231. i)T + 1.77e6iT^{2} \) |
| 17 | \( 1 + 7.52e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + (8.30e3 - 8.30e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 + 2.39e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 923.T + 5.94e8T^{2} \) |
| 31 | \( 1 + (1.64e4 - 1.64e4i)T - 8.87e8iT^{2} \) |
| 37 | \( 1 + (3.61e4 + 3.61e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + (-4.44e4 + 4.44e4i)T - 4.75e9iT^{2} \) |
| 43 | \( 1 + 9.79e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.12e5 - 1.12e5i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + 5.77e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + (5.74e4 + 5.74e4i)T + 4.21e10iT^{2} \) |
| 61 | \( 1 - 4.13e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + (2.65e5 - 2.65e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 + (2.88e4 - 2.88e4i)T - 1.28e11iT^{2} \) |
| 73 | \( 1 + (3.95e5 + 3.95e5i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 + 4.19e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-5.91e4 + 5.91e4i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + (6.10e4 + 6.10e4i)T + 4.96e11iT^{2} \) |
| 97 | \( 1 + (2.35e5 - 2.35e5i)T - 8.32e11iT^{2} \) |
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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
−17.62505308827414202313287668769, −16.97088676395882710618588950112, −16.15517354092305736856227346135, −14.36468916834217967267954458043, −12.46609348660127274248062268897, −10.26842134399211451059057481783, −9.002032996561932228357676248868, −7.14924713332811047125809759543, −5.61000219935485713789004375493, −0.24417087655973949509290138804,
2.48428543440956923572269466925, 6.18105194041926002579172058369, 8.814405883545936527889885504105, 10.18857586953912422307366210871, 11.32151525854577912889992329696, 12.63961590592940323807431184493, 14.80285168137269386252993258930, 16.96172770116906911843098648013, 17.67279156804013140637870721558, 19.00320391836345243122741647521