| L(s) = 1 | + (−4.41 − 9.16i)2-s + (−13.5 − 19.8i)3-s + (−24.6 + 30.8i)4-s + (24.4 − 26.3i)5-s + (−122. + 211. i)6-s + (−444. + 256. i)7-s + (−243. − 55.5i)8-s + (55.1 − 140. i)9-s + (−348. − 107. i)10-s + (738. + 926. i)11-s + (946. + 70.9i)12-s + (−1.12e3 + 346. i)13-s + (4.31e3 + 2.94e3i)14-s + (−853. − 128. i)15-s + (1.12e3 + 4.93e3i)16-s + (6.25e3 − 5.80e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.551 − 1.14i)2-s + (−0.501 − 0.735i)3-s + (−0.384 + 0.482i)4-s + (0.195 − 0.210i)5-s + (−0.566 + 0.980i)6-s + (−1.29 + 0.748i)7-s + (−0.475 − 0.108i)8-s + (0.0755 − 0.192i)9-s + (−0.348 − 0.107i)10-s + (0.555 + 0.696i)11-s + (0.547 + 0.0410i)12-s + (−0.511 + 0.157i)13-s + (1.57 + 1.07i)14-s + (−0.252 − 0.0381i)15-s + (0.275 + 1.20i)16-s + (1.27 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0632606 + 0.0280313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0632606 + 0.0280313i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.33e4 - 6.66e4i)T \) |
| good | 2 | \( 1 + (4.41 + 9.16i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (13.5 + 19.8i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (-24.4 + 26.3i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (444. - 256. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-738. - 926. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (1.12e3 - 346. i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (-6.25e3 + 5.80e3i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-709. + 278. i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (6.70e3 - 1.00e3i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (2.11e4 - 3.09e4i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (-1.91e3 + 2.55e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (3.01e4 + 1.73e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (809. - 389. i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (1.07e5 - 1.34e5i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (9.76e3 + 3.01e3i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (2.55e3 + 1.11e4i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (8.71e4 - 6.52e3i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (1.93e3 + 4.91e3i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (1.04e5 - 6.91e5i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (9.14e4 + 2.96e5i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (2.94e5 + 5.09e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-1.51e5 + 1.03e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (4.19e5 + 6.15e5i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (6.43e5 + 8.06e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.81621068194561697944717728959, −12.93292153509867559260056642112, −12.30844759787126421242676899587, −11.56201162682081403402140526228, −9.706561254620271928946501818995, −9.379555723079980326398211074399, −7.09669408221981854918695956524, −5.80694962544998461584556121946, −3.15532352633796915190769334845, −1.50404885274206901015672288228,
0.04189516129676920322588904197, 3.61818764724504501249284889721, 5.68866459429708517257084627239, 6.69032204285612023501830875453, 8.083098221464536339323453680526, 9.708724836143443294185275867588, 10.38478277639608458904779924577, 12.11966667381637193710765885334, 13.69491398810151153400987539729, 14.94668039554906720650052097679
