| L(s) = 1 | + (0.0550 − 0.258i)2-s + (−1.89 − 2.10i)3-s + (1.76 + 0.784i)4-s − 4.39i·5-s + (−0.649 + 0.374i)6-s + (0.553 − 1.24i)7-s + (0.611 − 0.841i)8-s + (−0.523 + 4.98i)9-s + (−1.13 − 0.241i)10-s + (−0.790 + 0.0830i)11-s + (−1.68 − 5.19i)12-s + (1.86 − 3.08i)13-s + (−0.291 − 0.211i)14-s + (−9.24 + 8.32i)15-s + (2.39 + 2.66i)16-s + (−0.311 + 2.95i)17-s + ⋯ |
| L(s) = 1 | + (0.0389 − 0.183i)2-s + (−1.09 − 1.21i)3-s + (0.881 + 0.392i)4-s − 1.96i·5-s + (−0.264 + 0.152i)6-s + (0.209 − 0.470i)7-s + (0.216 − 0.297i)8-s + (−0.174 + 1.66i)9-s + (−0.359 − 0.0764i)10-s + (−0.238 + 0.0250i)11-s + (−0.487 − 1.49i)12-s + (0.517 − 0.855i)13-s + (−0.0779 − 0.0566i)14-s + (−2.38 + 2.14i)15-s + (0.599 + 0.665i)16-s + (−0.0754 + 0.717i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.259323 - 1.14663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.259323 - 1.14663i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.86 + 3.08i)T \) |
| 31 | \( 1 + (-0.976 - 5.48i)T \) |
| good | 2 | \( 1 + (-0.0550 + 0.258i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (1.89 + 2.10i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + 4.39iT - 5T^{2} \) |
| 7 | \( 1 + (-0.553 + 1.24i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.790 - 0.0830i)T + (10.7 - 2.28i)T^{2} \) |
| 17 | \( 1 + (0.311 - 2.95i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-3.72 - 3.34i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.46 + 0.653i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (7.48 + 1.59i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + (1.72 + 0.995i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.48 + 11.6i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-1.12 + 1.25i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-1.63 - 0.532i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.74 - 3.44i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.926 - 4.36i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (0.0232 + 0.0402i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.35 - 5.40i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.79 + 0.188i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (1.68 + 2.31i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (1.29 + 0.938i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.8 + 3.53i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (16.8 - 1.76i)T + (87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (-2.37 + 5.33i)T + (-64.9 - 72.0i)T^{2} \) |
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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
−11.15333452025486978114285627334, −10.31314217748945548088982570806, −8.751124769635045860735992869563, −7.85213297227619428239061591649, −7.27723779367211267132005795733, −5.85307530268578331847605018746, −5.43133239951841017896860662572, −3.93052973269777978253619207190, −1.76892567222776342961440237761, −0.903268646275034468449414756038,
2.46846945322427321373799260104, 3.62297973572091945136946328228, 5.12031737228409064258452672294, 5.99673140808160064157953404633, 6.70250829268422301939582310004, 7.52768399484087369488347882995, 9.446980031100884177927282622312, 10.04569225005096281419503136672, 11.07716407422456158585782312287, 11.27733692640632735964784068832
