| L(s) = 1 | + (−0.264 + 0.163i)2-s + (−0.848 + 1.70i)4-s + (0.994 − 0.385i)5-s + (0.417 − 4.50i)7-s + (−0.112 − 1.20i)8-s + (−0.200 + 0.265i)10-s + (−4.48 + 2.77i)11-s + (0.414 − 4.47i)13-s + (0.628 + 1.26i)14-s + (−2.06 − 2.73i)16-s + (−5.95 + 5.42i)17-s + (−2.15 + 7.56i)19-s + (−0.187 + 2.02i)20-s + (0.732 − 1.47i)22-s + (1.19 + 0.740i)23-s + ⋯ |
| L(s) = 1 | + (−0.187 + 0.115i)2-s + (−0.424 + 0.851i)4-s + (0.444 − 0.172i)5-s + (0.157 − 1.70i)7-s + (−0.0396 − 0.427i)8-s + (−0.0632 + 0.0838i)10-s + (−1.35 + 0.837i)11-s + (0.114 − 1.24i)13-s + (0.167 + 0.337i)14-s + (−0.516 − 0.683i)16-s + (−1.44 + 1.31i)17-s + (−0.493 + 1.73i)19-s + (−0.0418 + 0.452i)20-s + (0.156 − 0.313i)22-s + (0.249 + 0.154i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000264597 - 0.110522i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000264597 - 0.110522i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 + (9.95 + 1.98i)T \) |
| good | 2 | \( 1 + (0.264 - 0.163i)T + (0.891 - 1.79i)T^{2} \) |
| 5 | \( 1 + (-0.994 + 0.385i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (-0.417 + 4.50i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (4.48 - 2.77i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (-0.414 + 4.47i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (5.95 - 5.42i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.15 - 7.56i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 0.740i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (0.0361 - 0.0139i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-0.548 - 0.726i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (4.98 + 0.931i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (2.48 + 0.961i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (1.49 - 0.278i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 1.62T + 47T^{2} \) |
| 53 | \( 1 + (2.55 - 8.96i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (1.00 + 10.8i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (-2.19 + 1.99i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (0.103 + 1.12i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (11.3 + 4.41i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (-8.31 - 3.21i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (-1.98 + 0.767i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (-0.571 + 6.16i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (0.471 + 0.946i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (-6.62 - 6.03i)T + (8.95 + 96.5i)T^{2} \) |
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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
−10.37509372331366814230473940530, −9.882139301474793814938573916363, −8.585975315861356483618402850006, −7.87580780466303585762063004580, −7.44656420891223804928679033034, −6.33200387755494288020990554527, −5.09494695454471936390546584067, −4.18519181114366831984913116081, −3.40589839860206614831088611679, −1.81858037789838238405531371441,
0.05140626339357539412918808479, 2.17594920767301577436494257921, 2.62171239072329372760441846157, 4.70701616882685895015997102291, 5.18659422974891869945577347762, 6.12428527536415089198862192492, 6.84666770155562807214488327888, 8.447553200399750758350578820578, 8.926388079494614352574066572617, 9.428624411986727442058392906828
