L(s) = 1 | + (2.82 − 4.36i)3-s + (9.22 + 15.9i)5-s + (6.70 − 17.2i)7-s + (−11.0 − 24.6i)9-s + (−11.8 − 6.86i)11-s + 9.57i·13-s + (95.7 + 4.87i)15-s + (48.2 − 83.5i)17-s + (23.2 − 13.4i)19-s + (−56.3 − 77.9i)21-s + (153. − 88.8i)23-s + (−107. + 186. i)25-s + (−138. − 21.2i)27-s + 131. i·29-s + (273. + 157. i)31-s + ⋯ |
L(s) = 1 | + (0.543 − 0.839i)3-s + (0.825 + 1.42i)5-s + (0.361 − 0.932i)7-s + (−0.409 − 0.912i)9-s + (−0.326 − 0.188i)11-s + 0.204i·13-s + (1.64 + 0.0838i)15-s + (0.688 − 1.19i)17-s + (0.280 − 0.162i)19-s + (−0.585 − 0.810i)21-s + (1.39 − 0.805i)23-s + (−0.862 + 1.49i)25-s + (−0.988 − 0.151i)27-s + 0.840i·29-s + (1.58 + 0.914i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.713571698\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.713571698\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.82 + 4.36i)T \) |
| 7 | \( 1 + (-6.70 + 17.2i)T \) |
good | 5 | \( 1 + (-9.22 - 15.9i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (11.8 + 6.86i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 9.57iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-48.2 + 83.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-23.2 + 13.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-153. + 88.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 131. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-273. - 157. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (54.7 + 94.7i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 390.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (149. + 258. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-6.68 - 3.85i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-133. + 231. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-195. + 113. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (296. - 512. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 864. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-474. - 274. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (91.1 + 157. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 54.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (403. + 698. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.42e3iT - 9.12e5T^{2} \) |
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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
−10.91200758341308622249721306849, −10.17709877508762452979297936323, −9.176394538023806146301827037285, −7.923161075117070889695467241834, −6.99053680423672259874771102643, −6.62022540326521010734560808985, −5.14559283284370040598000672710, −3.30998525525235686668836525412, −2.54925029059186940707667988446, −1.03915890268930721938987421686,
1.44578845149932664990505191359, 2.74925632190122225648248981286, 4.35250961792797111159481940066, 5.26321482722551291673834689665, 5.88679696392872617300063837758, 7.992305537805278555161360119569, 8.495533031555214072581472677242, 9.470273445118542219995150002986, 9.917915328037678791542128286215, 11.18695063871534345763493478599