| L(s) = 1 | + (7.51 − 2.73i)2-s + (37.5 + 27.9i)3-s + (49.0 − 41.1i)4-s + (−21.4 − 121. i)5-s + (358. + 107. i)6-s + (674. + 565. i)7-s + (256. − 443. i)8-s + (629. + 2.09e3i)9-s + (−494. − 856. i)10-s + (854. − 4.84e3i)11-s + (2.98e3 − 175. i)12-s + (8.39e3 + 3.05e3i)13-s + (6.61e3 + 2.40e3i)14-s + (2.59e3 − 5.16e3i)15-s + (711. − 4.03e3i)16-s + (2.81e3 + 4.88e3i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.802 + 0.596i)3-s + (0.383 − 0.321i)4-s + (−0.0767 − 0.435i)5-s + (0.677 + 0.202i)6-s + (0.743 + 0.623i)7-s + (0.176 − 0.306i)8-s + (0.287 + 0.957i)9-s + (−0.156 − 0.270i)10-s + (0.193 − 1.09i)11-s + (0.499 − 0.0293i)12-s + (1.06 + 0.385i)13-s + (0.644 + 0.234i)14-s + (0.198 − 0.395i)15-s + (0.0434 − 0.246i)16-s + (0.139 + 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.93728 + 0.232903i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.93728 + 0.232903i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.51 + 2.73i)T \) |
| 3 | \( 1 + (-37.5 - 27.9i)T \) |
| good | 5 | \( 1 + (21.4 + 121. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-674. - 565. i)T + (1.43e5 + 8.11e5i)T^{2} \) |
| 11 | \( 1 + (-854. + 4.84e3i)T + (-1.83e7 - 6.66e6i)T^{2} \) |
| 13 | \( 1 + (-8.39e3 - 3.05e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-2.81e3 - 4.88e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.59e4 - 2.75e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.27e4 + 1.07e4i)T + (5.91e8 - 3.35e9i)T^{2} \) |
| 29 | \( 1 + (1.97e4 - 7.18e3i)T + (1.32e10 - 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-1.06e5 + 8.95e4i)T + (4.77e9 - 2.70e10i)T^{2} \) |
| 37 | \( 1 + (1.79e5 + 3.11e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (3.56e5 + 1.29e5i)T + (1.49e11 + 1.25e11i)T^{2} \) |
| 43 | \( 1 + (1.33e5 - 7.56e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (1.61e5 + 1.35e5i)T + (8.79e10 + 4.98e11i)T^{2} \) |
| 53 | \( 1 + 1.41e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-4.82e4 - 2.73e5i)T + (-2.33e12 + 8.51e11i)T^{2} \) |
| 61 | \( 1 + (1.20e6 + 1.01e6i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (3.54e6 + 1.29e6i)T + (4.64e12 + 3.89e12i)T^{2} \) |
| 71 | \( 1 + (1.63e6 + 2.82e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.60e6 - 4.51e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (5.50e6 - 2.00e6i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-4.50e6 + 1.64e6i)T + (2.07e13 - 1.74e13i)T^{2} \) |
| 89 | \( 1 + (-5.12e6 + 8.88e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (5.91e5 - 3.35e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−14.02635060900110344812484727389, −12.94154943154023157097090064311, −11.56380045048167224031169004158, −10.56733625550179027774590093817, −8.929459537471588635278844038416, −8.180129619285511830652846938129, −5.98672553874172685492004478485, −4.59799123930959605236190642355, −3.32864392701431517230078013432, −1.64590839086306980463551260776,
1.49696269255512845683171717630, 3.14695249326808788333524613224, 4.60325468028951629660523356637, 6.62367693031396903769911776749, 7.49288848137919931006270065106, 8.726928380780324291292485526201, 10.48720312109808068022794094436, 11.78419934644478579447047480558, 13.04196208905797946513048955134, 13.85228077528096144386789224676
