| L(s) = 1 | + (0.772 − 0.561i)2-s + (0.927 − 2.85i)3-s + (−2.19 + 6.74i)4-s + (11.0 + 1.36i)5-s + (−0.885 − 2.72i)6-s + 12.4·7-s + (4.45 + 13.7i)8-s + (−7.28 − 5.29i)9-s + (9.34 − 5.17i)10-s + (36.1 − 26.2i)11-s + (17.2 + 12.4i)12-s + (5.77 + 4.19i)13-s + (9.64 − 7.00i)14-s + (14.1 − 30.3i)15-s + (−34.7 − 25.2i)16-s + (8.19 + 25.2i)17-s + ⋯ |
| L(s) = 1 | + (0.273 − 0.198i)2-s + (0.178 − 0.549i)3-s + (−0.273 + 0.842i)4-s + (0.992 + 0.121i)5-s + (−0.0602 − 0.185i)6-s + 0.674·7-s + (0.196 + 0.605i)8-s + (−0.269 − 0.195i)9-s + (0.295 − 0.163i)10-s + (0.991 − 0.720i)11-s + (0.413 + 0.300i)12-s + (0.123 + 0.0894i)13-s + (0.184 − 0.133i)14-s + (0.243 − 0.523i)15-s + (−0.542 − 0.394i)16-s + (0.116 + 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.04214 - 0.121259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04214 - 0.121259i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.927 + 2.85i)T \) |
| 5 | \( 1 + (-11.0 - 1.36i)T \) |
| good | 2 | \( 1 + (-0.772 + 0.561i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 - 12.4T + 343T^{2} \) |
| 11 | \( 1 + (-36.1 + 26.2i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-5.77 - 4.19i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-8.19 - 25.2i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (14.3 + 44.1i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (117. - 85.4i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (65.9 - 202. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (45.7 + 140. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (325. + 236. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (189. + 137. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 87.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + (23.4 - 72.2i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-51.6 + 158. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-481. - 349. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-700. + 508. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (125. + 386. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-162. + 500. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (810. - 588. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (41.3 - 127. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-159. - 491. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-494. + 358. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (506. - 1.55e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.91930413913783210188992834532, −13.06751099221860337323655106200, −11.97053877402226874942046021543, −10.97590962288203084098525657150, −9.241572345699803094805104527591, −8.343009579516635200700247275704, −6.95053586190966358408470878303, −5.48429889627171868306159403061, −3.65480270348183813927669542795, −1.88821151870874401477082686390,
1.72577984213063747540763963961, 4.29716119814912819217591622670, 5.41788675856106908711062131465, 6.61916590030207771112799161113, 8.588728731007406966253141437088, 9.742553129174834703710774684550, 10.35476399412287442623517673263, 11.85894671946407242669944099832, 13.35279550828983590573605517957, 14.34855441380048536944983318308
