| L(s) = 1 | − 5.28i·2-s + (3.72 − 2.15i)3-s − 11.9·4-s + (30.9 − 17.8i)5-s + (−11.3 − 19.7i)6-s + (−25.0 − 14.4i)7-s − 21.3i·8-s + (−31.2 + 54.1i)9-s + (−94.4 − 163. i)10-s + 53.2·11-s + (−44.6 + 25.7i)12-s + (−145. + 252. i)13-s + (−76.3 + 132. i)14-s + (76.8 − 133. i)15-s − 304.·16-s + (215. − 373. i)17-s + ⋯ |
| L(s) = 1 | − 1.32i·2-s + (0.414 − 0.239i)3-s − 0.748·4-s + (1.23 − 0.714i)5-s + (−0.316 − 0.547i)6-s + (−0.510 − 0.294i)7-s − 0.332i·8-s + (−0.385 + 0.668i)9-s + (−0.944 − 1.63i)10-s + 0.440·11-s + (−0.309 + 0.178i)12-s + (−0.860 + 1.49i)13-s + (−0.389 + 0.674i)14-s + (0.341 − 0.591i)15-s − 1.18·16-s + (0.746 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.774677 - 1.71921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.774677 - 1.71921i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.27e3 - 1.33e3i)T \) |
| good | 2 | \( 1 + 5.28iT - 16T^{2} \) |
| 3 | \( 1 + (-3.72 + 2.15i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-30.9 + 17.8i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (25.0 + 14.4i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 53.2T + 1.46e4T^{2} \) |
| 13 | \( 1 + (145. - 252. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + (-215. + 373. i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-403. + 232. i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-308. - 534. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-52.7 - 30.4i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-358. - 620. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-476. + 275. i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 861.T + 2.82e6T^{2} \) |
| 47 | \( 1 + 235.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-525. - 909. i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + 4.97e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + (5.24e3 + 3.02e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-165. - 286. i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-2.21e3 - 1.28e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-7.57e3 - 4.37e3i)T + (1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (1.25e3 - 2.17e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (476. + 825. i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (9.11e3 - 5.26e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + 8.50e3T + 8.85e7T^{2} \) |
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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
−13.92336661391745914058735411366, −13.66787824547270467776055507470, −12.35955407888224510196786574282, −11.30078806253682591327090303856, −9.549034149031977424644784075254, −9.409553144813469205896346618872, −7.05719249058547033277298947844, −4.99981092039565951562493047615, −2.83585963768433464860748026789, −1.42690539786634983143137739238,
2.91757573455006477307633033953, 5.64705414059038027875638306142, 6.37676133176928553050553829847, 7.916271328294922097856289499854, 9.345819687220992833867886997657, 10.35008493088816430104348790827, 12.39884193050613088784622258253, 13.89708880594937553689195991832, 14.74192649592159044684062919102, 15.27494508238908823438397157759
