| L(s) = 1 | + (−1.38 + 7.87i)2-s + (−12.0 − 10.1i)3-s + (−60.1 − 21.8i)4-s + (−307. + 111. i)5-s + (96.6 − 81.1i)6-s + (444. + 769. i)7-s + (256 − 443. i)8-s + (−336. − 1.90e3i)9-s + (−454. − 2.57e3i)10-s + (1.66e3 − 2.88e3i)11-s + (504. + 874. i)12-s + (2.83e3 − 2.38e3i)13-s + (−6.67e3 + 2.43e3i)14-s + (4.84e3 + 1.76e3i)15-s + (3.13e3 + 2.63e3i)16-s + (3.20e3 − 1.81e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.258 − 0.216i)3-s + (−0.469 − 0.171i)4-s + (−1.09 + 0.400i)5-s + (0.182 − 0.153i)6-s + (0.489 + 0.847i)7-s + (0.176 − 0.306i)8-s + (−0.153 − 0.872i)9-s + (−0.143 − 0.815i)10-s + (0.377 − 0.653i)11-s + (0.0843 + 0.146i)12-s + (0.358 − 0.300i)13-s + (−0.650 + 0.236i)14-s + (0.371 + 0.135i)15-s + (0.191 + 0.160i)16-s + (0.158 − 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.920123 - 0.288291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.920123 - 0.288291i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 - 7.87i)T \) |
| 19 | \( 1 + (-2.68e4 + 1.31e4i)T \) |
| good | 3 | \( 1 + (12.0 + 10.1i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (307. - 111. i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-444. - 769. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.66e3 + 2.88e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-2.83e3 + 2.38e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-3.20e3 + 1.81e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-3.95e4 - 1.43e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (3.13e4 + 1.77e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (8.98e3 + 1.55e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 2.10e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.60e5 + 3.02e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (6.53e5 - 2.37e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (9.55e4 + 5.41e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (-1.86e6 - 6.77e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-3.59e5 + 2.03e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (2.21e6 + 8.07e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (3.66e5 + 2.07e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-3.72e6 + 1.35e6i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-2.93e6 - 2.46e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-1.08e6 - 9.09e5i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-4.80e5 - 8.32e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.97e6 - 1.65e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (1.24e6 - 7.04e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−15.09553321425012956602718374556, −13.70148067609729510944735723907, −11.96445462854736084122241246982, −11.36308525622072344850672998385, −9.293246506633307358054271199878, −8.107181142601836814480103067453, −6.83369564308744442876110602456, −5.40443928442491319578346807134, −3.45612533089147015350280960701, −0.53079530270186799426691975463,
1.33796943760112014987839951051, 3.79323673245723826205529354660, 4.89359476105209420931330491549, 7.41641924678330643018523858620, 8.534961765747438160894609455606, 10.26887009326920892430966955229, 11.23321284911812788822611326430, 12.20468970499572016783350210986, 13.54001312186636468337039741203, 14.83199624565326201182422180852
