| L(s) = 1 | + (−2.26 + 12.8i)5-s + (−3.04 + 2.55i)7-s + (1.56 + 8.85i)11-s + (−33.8 + 12.3i)13-s + (−0.988 + 1.71i)17-s + (−56.3 − 97.5i)19-s + (73.1 + 61.3i)23-s + (−42.2 − 15.3i)25-s + (−237. − 86.3i)29-s + (−154. − 129. i)31-s + (−25.8 − 44.8i)35-s + (112. − 195. i)37-s + (−209. + 76.3i)41-s + (−10.6 − 60.5i)43-s + (−140. + 118. i)47-s + ⋯ |
| L(s) = 1 | + (−0.202 + 1.14i)5-s + (−0.164 + 0.137i)7-s + (0.0427 + 0.242i)11-s + (−0.722 + 0.263i)13-s + (−0.0141 + 0.0244i)17-s + (−0.679 − 1.17i)19-s + (0.662 + 0.556i)23-s + (−0.337 − 0.122i)25-s + (−1.51 − 0.553i)29-s + (−0.894 − 0.750i)31-s + (−0.124 − 0.216i)35-s + (0.500 − 0.866i)37-s + (−0.799 + 0.290i)41-s + (−0.0378 − 0.214i)43-s + (−0.437 + 0.366i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3065676497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3065676497\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2.26 - 12.8i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (3.04 - 2.55i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (-1.56 - 8.85i)T + (-1.25e3 + 455. i)T^{2} \) |
| 13 | \( 1 + (33.8 - 12.3i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (0.988 - 1.71i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (56.3 + 97.5i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-73.1 - 61.3i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (237. + 86.3i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (154. + 129. i)T + (5.17e3 + 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-112. + 195. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (209. - 76.3i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (10.6 + 60.5i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (140. - 118. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + 596.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (26.4 - 149. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (282. - 236. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-869. + 316. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (384. - 666. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (329. + 570. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (1.09e3 + 400. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-189. - 68.7i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-246. - 427. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (95.1 + 539. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−11.34769003149404341417174414415, −11.01028847786512395996988923940, −9.793969458988091947682107090722, −9.060754370363774264367364427776, −7.58502874010534459430647297012, −7.01871322043568304255863231765, −5.94549503007746538341776145914, −4.58693454686257469885340611165, −3.28432475149150016547735191490, −2.17424595812837452914619277450,
0.10570171492798144062113731643, 1.63739416365699863252409779751, 3.41306274125346222008058256540, 4.65416237034119429937851029361, 5.52246540752644310072829133441, 6.82169207598525474255851658274, 7.983467190414104113527872084201, 8.743732474752460670165306223636, 9.668447945036515713819910243638, 10.65509418216685089396873542489
