| L(s) = 1 | + (−0.540 + 0.841i)2-s + (−1.59 + 0.229i)3-s + (−0.415 − 0.909i)4-s + (0.883 + 2.05i)5-s + (0.671 − 1.46i)6-s + (−1.35 − 1.17i)7-s + (0.989 + 0.142i)8-s + (−0.374 + 0.109i)9-s + (−2.20 − 0.366i)10-s + (−4.81 + 3.09i)11-s + (0.873 + 1.35i)12-s + (1.68 − 1.45i)13-s + (1.71 − 0.504i)14-s + (−1.88 − 3.08i)15-s + (−0.654 + 0.755i)16-s + (−3.28 − 1.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 + 0.594i)2-s + (−0.923 + 0.132i)3-s + (−0.207 − 0.454i)4-s + (0.395 + 0.918i)5-s + (0.273 − 0.599i)6-s + (−0.511 − 0.442i)7-s + (0.349 + 0.0503i)8-s + (−0.124 + 0.0366i)9-s + (−0.697 − 0.116i)10-s + (−1.45 + 0.932i)11-s + (0.252 + 0.392i)12-s + (0.466 − 0.404i)13-s + (0.458 − 0.134i)14-s + (−0.486 − 0.795i)15-s + (−0.163 + 0.188i)16-s + (−0.797 − 0.364i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0422642 - 0.213572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0422642 - 0.213572i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (-0.883 - 2.05i)T \) |
| 23 | \( 1 + (4.17 - 2.36i)T \) |
| good | 3 | \( 1 + (1.59 - 0.229i)T + (2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (1.35 + 1.17i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (4.81 - 3.09i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.68 + 1.45i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.28 + 1.50i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (3.15 + 6.90i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.955 - 2.09i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.29 - 9.02i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.17 - 3.99i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.64 - 1.65i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-2.25 + 0.324i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 3.81iT - 47T^{2} \) |
| 53 | \( 1 + (-2.96 - 2.57i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.34 - 7.32i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.721 - 5.01i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.00 + 3.12i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (0.655 + 0.421i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (13.8 - 6.32i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (6.94 + 8.01i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (4.32 + 14.7i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.0331 - 0.230i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.29 - 11.2i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−12.94314352343747230877465315043, −11.44051971406131332530230809714, −10.55860011851821647959713808143, −10.20563642455313545789275521271, −8.870224338853607418614726407367, −7.45111277481664735723005105647, −6.72175848567906903073332288907, −5.74885498648332823881927829953, −4.70805385210310539975909073814, −2.67613378200948172776763938066,
0.20899647138042545927270518474, 2.26094217927654981566899443690, 4.10345109503634920722987529238, 5.68038506459254831193704325834, 6.08386026784156753400500731800, 8.060973692052219718171505366012, 8.727534621987154407287496051228, 9.883365124222143405694680469744, 10.79762719696573622953544712365, 11.59925524399849097453293722749
