| L(s) = 1 | + (−15.0 + 5.47i)2-s + (48.1 − 272. i)3-s + (196. − 164. i)4-s + (−1.03e3 − 866. i)5-s + (769. + 4.36e3i)6-s + (−4.22e3 − 7.31e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (−5.36e4 − 1.95e4i)9-s + (2.02e4 + 7.37e3i)10-s + (2.23e4 − 3.86e4i)11-s + (−3.54e4 − 6.14e4i)12-s + (2.57e3 + 1.45e4i)13-s + (1.03e5 + 8.68e4i)14-s + (−2.86e5 + 2.40e5i)15-s + (1.13e4 − 6.45e4i)16-s + (5.31e5 − 1.93e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.342 − 1.94i)3-s + (0.383 − 0.321i)4-s + (−0.739 − 0.620i)5-s + (0.242 + 1.37i)6-s + (−0.664 − 1.15i)7-s + (−0.176 + 0.306i)8-s + (−2.72 − 0.991i)9-s + (0.641 + 0.233i)10-s + (0.459 − 0.795i)11-s + (−0.493 − 0.854i)12-s + (0.0249 + 0.141i)13-s + (0.719 + 0.604i)14-s + (−1.45 + 1.22i)15-s + (0.0434 − 0.246i)16-s + (1.54 − 0.561i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.456759 + 0.881717i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456759 + 0.881717i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.0 - 5.47i)T \) |
| 19 | \( 1 + (-5.53e5 + 1.27e5i)T \) |
| good | 3 | \( 1 + (-48.1 + 272. i)T + (-1.84e4 - 6.73e3i)T^{2} \) |
| 5 | \( 1 + (1.03e3 + 866. i)T + (3.39e5 + 1.92e6i)T^{2} \) |
| 7 | \( 1 + (4.22e3 + 7.31e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.23e4 + 3.86e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-2.57e3 - 1.45e4i)T + (-9.96e9 + 3.62e9i)T^{2} \) |
| 17 | \( 1 + (-5.31e5 + 1.93e5i)T + (9.08e10 - 7.62e10i)T^{2} \) |
| 23 | \( 1 + (-1.97e5 + 1.65e5i)T + (3.12e11 - 1.77e12i)T^{2} \) |
| 29 | \( 1 + (2.67e6 + 9.74e5i)T + (1.11e13 + 9.32e12i)T^{2} \) |
| 31 | \( 1 + (-4.84e5 - 8.38e5i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.96e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-3.14e5 + 1.78e6i)T + (-3.07e14 - 1.11e14i)T^{2} \) |
| 43 | \( 1 + (-1.46e7 - 1.23e7i)T + (8.72e13 + 4.94e14i)T^{2} \) |
| 47 | \( 1 + (-5.24e6 - 1.90e6i)T + (8.57e14 + 7.19e14i)T^{2} \) |
| 53 | \( 1 + (1.47e7 - 1.23e7i)T + (5.72e14 - 3.24e15i)T^{2} \) |
| 59 | \( 1 + (1.43e8 - 5.23e7i)T + (6.63e15 - 5.56e15i)T^{2} \) |
| 61 | \( 1 + (-8.15e7 + 6.83e7i)T + (2.03e15 - 1.15e16i)T^{2} \) |
| 67 | \( 1 + (1.02e8 + 3.72e7i)T + (2.08e16 + 1.74e16i)T^{2} \) |
| 71 | \( 1 + (-1.71e8 - 1.44e8i)T + (7.96e15 + 4.51e16i)T^{2} \) |
| 73 | \( 1 + (-6.78e7 + 3.84e8i)T + (-5.53e16 - 2.01e16i)T^{2} \) |
| 79 | \( 1 + (-3.96e7 + 2.25e8i)T + (-1.12e17 - 4.09e16i)T^{2} \) |
| 83 | \( 1 + (5.77e7 + 9.99e7i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (4.82e7 + 2.73e8i)T + (-3.29e17 + 1.19e17i)T^{2} \) |
| 97 | \( 1 + (-6.40e8 + 2.33e8i)T + (5.82e17 - 4.88e17i)T^{2} \) |
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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
−13.50892782081143362371177402066, −12.33469856565608916201933054352, −11.38573310704724842101191060263, −9.306557418129867276694040469782, −7.934487600103114404683150820450, −7.36764149528694138839204964580, −6.08883411386153409370463662667, −3.21048750559204806076798089379, −1.04811113473462145205505124877, −0.54726283409259755676165222208,
2.86274935768955842320158267765, 3.78140776604508274845337373518, 5.63282231363864235527815747209, 7.88579405691862412808707369993, 9.295425947463123052243937003221, 9.871330339355440259617620931378, 11.12517659934800319799824944887, 12.14634332137548372134528392683, 14.59697604274105720236091395132, 15.23348847863657048659680169729
