| L(s) = 1 | + (−0.100 + 1.84i)2-s + (2.00 − 4.79i)3-s + (4.56 + 0.495i)4-s + (6.12 − 18.1i)5-s + (8.63 + 4.18i)6-s + (1.32 + 1.95i)7-s + (−3.76 + 22.9i)8-s + (−18.9 − 19.2i)9-s + (32.8 + 13.1i)10-s + (34.4 − 7.58i)11-s + (11.5 − 20.8i)12-s + (−2.61 + 2.22i)13-s + (−3.73 + 2.24i)14-s + (−74.7 − 65.8i)15-s + (−6.11 − 1.34i)16-s + (−95.2 − 64.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0353 + 0.652i)2-s + (0.386 − 0.922i)3-s + (0.570 + 0.0619i)4-s + (0.547 − 1.62i)5-s + (0.587 + 0.284i)6-s + (0.0715 + 0.105i)7-s + (−0.166 + 1.01i)8-s + (−0.701 − 0.713i)9-s + (1.04 + 0.414i)10-s + (0.944 − 0.207i)11-s + (0.277 − 0.501i)12-s + (−0.0557 + 0.0473i)13-s + (−0.0713 + 0.0429i)14-s + (−1.28 − 1.13i)15-s + (−0.0954 − 0.0210i)16-s + (−1.35 − 0.921i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.11377 - 1.13128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11377 - 1.13128i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.00 + 4.79i)T \) |
| 59 | \( 1 + (-149. - 427. i)T \) |
| good | 2 | \( 1 + (0.100 - 1.84i)T + (-7.95 - 0.864i)T^{2} \) |
| 5 | \( 1 + (-6.12 + 18.1i)T + (-99.5 - 75.6i)T^{2} \) |
| 7 | \( 1 + (-1.32 - 1.95i)T + (-126. + 318. i)T^{2} \) |
| 11 | \( 1 + (-34.4 + 7.58i)T + (1.20e3 - 558. i)T^{2} \) |
| 13 | \( 1 + (2.61 - 2.22i)T + (355. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (95.2 + 64.5i)T + (1.81e3 + 4.56e3i)T^{2} \) |
| 19 | \( 1 + (31.0 - 58.5i)T + (-3.84e3 - 5.67e3i)T^{2} \) |
| 23 | \( 1 + (26.8 - 25.3i)T + (658. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-269. + 14.6i)T + (2.42e4 - 2.63e3i)T^{2} \) |
| 31 | \( 1 + (-285. + 151. i)T + (1.67e4 - 2.46e4i)T^{2} \) |
| 37 | \( 1 + (-217. + 35.6i)T + (4.80e4 - 1.61e4i)T^{2} \) |
| 41 | \( 1 + (272. - 287. i)T + (-3.73e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (56.7 - 257. i)T + (-7.21e4 - 3.33e4i)T^{2} \) |
| 47 | \( 1 + (70.1 - 23.6i)T + (8.26e4 - 6.28e4i)T^{2} \) |
| 53 | \( 1 + (-170. + 67.7i)T + (1.08e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (126. + 6.84i)T + (2.25e5 + 2.45e4i)T^{2} \) |
| 67 | \( 1 + (-218. - 35.7i)T + (2.85e5 + 9.60e4i)T^{2} \) |
| 71 | \( 1 + (-181. - 537. i)T + (-2.84e5 + 2.16e5i)T^{2} \) |
| 73 | \( 1 + (361. + 601. i)T + (-1.82e5 + 3.43e5i)T^{2} \) |
| 79 | \( 1 + (70.0 + 32.4i)T + (3.19e5 + 3.75e5i)T^{2} \) |
| 83 | \( 1 + (-272. - 983. i)T + (-4.89e5 + 2.94e5i)T^{2} \) |
| 89 | \( 1 + (59.0 + 1.09e3i)T + (-7.00e5 + 7.62e4i)T^{2} \) |
| 97 | \( 1 + (660. - 1.09e3i)T + (-4.27e5 - 8.06e5i)T^{2} \) |
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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
−12.02319041423634434155493702625, −11.64008631699506143142695561017, −9.667460837821946884071456184467, −8.616745673245131480868440897658, −8.144492467936382797561907550117, −6.68593335306466180396437043712, −6.01354489763305188355381738719, −4.62477833052027891863419275208, −2.39556132657993024412440701105, −1.11028358660289812252788771328,
2.15828292021634218687252836592, 3.05891066299803932194827681601, 4.27945725405505166749634000520, 6.31615292544332821229290021580, 6.86169412574689512380227323776, 8.594849681207238966241594040966, 9.845734748180466348977466053178, 10.49871479888784451078778903445, 11.03399656013491340785129937854, 12.01118151630179642890850442080
