L(s) = 1 | + (7.51 − 2.73i)2-s + (−3.06 + 17.4i)3-s + (49.0 − 41.1i)4-s + (−9.97 − 8.37i)5-s + (24.5 + 139. i)6-s + (364. + 631. i)7-s + (256. − 443. i)8-s + (1.76e3 + 641. i)9-s + (−97.9 − 35.6i)10-s + (2.19e3 − 3.79e3i)11-s + (565. + 979. i)12-s + (2.25e3 + 1.27e4i)13-s + (4.47e3 + 3.75e3i)14-s + (176. − 147. i)15-s + (711. − 4.03e3i)16-s + (1.90e4 − 6.94e3i)17-s + ⋯ |
L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.0656 + 0.372i)3-s + (0.383 − 0.321i)4-s + (−0.0356 − 0.0299i)5-s + (0.0464 + 0.263i)6-s + (0.401 + 0.696i)7-s + (0.176 − 0.306i)8-s + (0.805 + 0.293i)9-s + (−0.0309 − 0.0112i)10-s + (0.496 − 0.859i)11-s + (0.0945 + 0.163i)12-s + (0.284 + 1.61i)13-s + (0.435 + 0.365i)14-s + (0.0134 − 0.0113i)15-s + (0.0434 − 0.246i)16-s + (0.941 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.81088 + 0.474469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.81088 + 0.474469i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-7.51 + 2.73i)T \) |
| 19 | \( 1 + (1.60e3 - 2.98e4i)T \) |
good | 3 | \( 1 + (3.06 - 17.4i)T + (-2.05e3 - 747. i)T^{2} \) |
| 5 | \( 1 + (9.97 + 8.37i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (-364. - 631. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-2.19e3 + 3.79e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-2.25e3 - 1.27e4i)T + (-5.89e7 + 2.14e7i)T^{2} \) |
| 17 | \( 1 + (-1.90e4 + 6.94e3i)T + (3.14e8 - 2.63e8i)T^{2} \) |
| 23 | \( 1 + (-4.64e4 + 3.89e4i)T + (5.91e8 - 3.35e9i)T^{2} \) |
| 29 | \( 1 + (9.54e4 + 3.47e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-7.62e4 - 1.32e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 4.86e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.74e3 - 2.12e4i)T + (-1.83e11 - 6.66e10i)T^{2} \) |
| 43 | \( 1 + (5.38e5 + 4.51e5i)T + (4.72e10 + 2.67e11i)T^{2} \) |
| 47 | \( 1 + (9.61e5 + 3.49e5i)T + (3.88e11 + 3.25e11i)T^{2} \) |
| 53 | \( 1 + (7.88e5 - 6.61e5i)T + (2.03e11 - 1.15e12i)T^{2} \) |
| 59 | \( 1 + (-2.38e6 + 8.68e5i)T + (1.90e12 - 1.59e12i)T^{2} \) |
| 61 | \( 1 + (3.67e4 - 3.08e4i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-4.29e6 - 1.56e6i)T + (4.64e12 + 3.89e12i)T^{2} \) |
| 71 | \( 1 + (6.73e5 + 5.64e5i)T + (1.57e12 + 8.95e12i)T^{2} \) |
| 73 | \( 1 + (-1.38e5 + 7.84e5i)T + (-1.03e13 - 3.77e12i)T^{2} \) |
| 79 | \( 1 + (-4.39e5 + 2.49e6i)T + (-1.80e13 - 6.56e12i)T^{2} \) |
| 83 | \( 1 + (2.46e5 + 4.26e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (5.41e5 + 3.07e6i)T + (-4.15e13 + 1.51e13i)T^{2} \) |
| 97 | \( 1 + (1.07e7 - 3.91e6i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−14.64516888775079525028719910841, −13.81678999004966386883330442257, −12.29270595973487308321003806215, −11.41858845871844721779933682831, −10.05053717362054349837576013405, −8.596093469898054402827603820442, −6.68004579095673978756223811865, −5.14390801244430044507358085062, −3.74905613140459284802351946599, −1.69627309397656143317247877959,
1.28346777083983728620284865109, 3.57967674147947348768156145987, 5.15735477500425948267566640359, 6.88978058478167019177505419337, 7.82187743463170757362242581619, 9.868549520507272285162877935669, 11.24954555846202462761850555679, 12.65911587520577849089361516526, 13.33413054032538993803642234279, 14.81923920921539852223600877755