| L(s) = 1 | + (−0.0747 − 0.997i)2-s + (1.08 + 0.334i)3-s + (−0.988 + 0.149i)4-s + (−2.80 + 2.60i)5-s + (0.252 − 1.10i)6-s + (0.222 + 0.974i)8-s + (−1.41 − 0.965i)9-s + (2.80 + 2.60i)10-s + (2.00 − 1.36i)11-s + (−1.12 − 0.168i)12-s + (−3.52 − 1.69i)13-s + (−3.90 + 1.88i)15-s + (0.955 − 0.294i)16-s + (−1.90 − 4.86i)17-s + (−0.857 + 1.48i)18-s + (−1.60 − 2.77i)19-s + ⋯ |
| L(s) = 1 | + (−0.0528 − 0.705i)2-s + (0.625 + 0.192i)3-s + (−0.494 + 0.0745i)4-s + (−1.25 + 1.16i)5-s + (0.102 − 0.451i)6-s + (0.0786 + 0.344i)8-s + (−0.472 − 0.321i)9-s + (0.887 + 0.823i)10-s + (0.604 − 0.411i)11-s + (−0.323 − 0.0487i)12-s + (−0.977 − 0.470i)13-s + (−1.00 + 0.486i)15-s + (0.238 − 0.0736i)16-s + (−0.462 − 1.17i)17-s + (−0.202 + 0.350i)18-s + (−0.367 − 0.636i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00163457 + 0.223966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00163457 + 0.223966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0747 + 0.997i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-1.08 - 0.334i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (2.80 - 2.60i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-2.00 + 1.36i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (3.52 + 1.69i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (1.90 + 4.86i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (1.60 + 2.77i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.50 - 3.82i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (0.332 - 0.416i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (3.70 - 6.42i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (11.5 + 1.73i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (0.288 + 1.26i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.0321 + 0.140i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.221 + 2.95i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-5.40 + 0.815i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-6.24 - 5.79i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-2.28 - 0.344i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (3.95 - 6.85i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.76 - 7.23i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.296 + 3.95i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (6.52 + 11.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.622 + 0.299i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-2.01 - 1.37i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−10.18646265741041198231626806766, −9.137169892602378150862921907388, −8.517663721440553282887619232141, −7.42246663207862247557563486947, −6.84100152775726866722973970644, −5.26688208438980271748489087633, −3.95642658838077848445790407999, −3.27674644163049280390030616329, −2.51113304424762896451026817814, −0.10656078228696568562363922348,
1.91054788564833990858574261864, 3.76912840404776338287703674112, 4.42021547888960462908154270344, 5.42452284816197691219277938135, 6.71928596741423641354016434345, 7.65884531115320948680832708320, 8.314499587770216909099609200794, 8.789663113374509595084713291846, 9.680810302909836997881663984434, 10.94261231749534332562455805599
