| L(s) = 1 | + (−0.931 + 2.13i)2-s + (−2.33 − 2.51i)4-s + (2.63 + 0.396i)5-s + (−2.74 − 2.54i)7-s + (3.14 − 1.10i)8-s + (−3.30 + 5.25i)10-s + (0.317 − 0.273i)11-s + (0.316 − 0.463i)13-s + (8.00 − 3.49i)14-s + (−0.0678 + 0.905i)16-s + (−2.10 − 2.10i)17-s + (−3.27 − 2.05i)19-s + (−5.14 − 7.54i)20-s + (0.287 + 0.932i)22-s + (−0.386 − 0.151i)23-s + ⋯ |
| L(s) = 1 | + (−0.658 + 1.51i)2-s + (−1.16 − 1.25i)4-s + (1.17 + 0.177i)5-s + (−1.03 − 0.963i)7-s + (1.11 − 0.389i)8-s + (−1.04 + 1.66i)10-s + (0.0957 − 0.0823i)11-s + (0.0876 − 0.128i)13-s + (2.13 − 0.933i)14-s + (−0.0169 + 0.226i)16-s + (−0.510 − 0.510i)17-s + (−0.750 − 0.471i)19-s + (−1.15 − 1.68i)20-s + (0.0613 + 0.198i)22-s + (−0.0805 − 0.0316i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.843483 + 0.0373773i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.843483 + 0.0373773i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (-4.54 + 2.89i)T \) |
| good | 2 | \( 1 + (0.931 - 2.13i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (-2.63 - 0.396i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (2.74 + 2.54i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.317 + 0.273i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.316 + 0.463i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (2.10 + 2.10i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.27 + 2.05i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (0.386 + 0.151i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-8.11 + 5.98i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (0.872 + 2.49i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (8.79 + 2.35i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.97 + 9.45i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (3.52 + 4.09i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-10.7 - 8.53i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (-4.46 + 2.57i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (13.3 - 0.498i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (2.39 - 0.179i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-7.83 + 3.77i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.882 - 7.83i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-1.19 + 6.29i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (2.83 - 9.18i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (3.90 - 0.439i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-2.11 - 3.99i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−10.08010161043603775072052414822, −9.334511288488210175035963703631, −8.622743971910566977313102019255, −7.53861164916233384062456754345, −6.64408340722306544413563006149, −6.36007604594765384597489001340, −5.38851115429741366775062695669, −4.17522572933486541314154852914, −2.55334105329171041767018922833, −0.53569722854808494025337236095,
1.48043086295024113793666871101, 2.45028827975787214246977083802, 3.26174779139245727141079088607, 4.64052093860918006330332986696, 6.00784454453449539795021691086, 6.54887985788509795678903622514, 8.425588697533788935860601882189, 8.819594797088771101325613799242, 9.780148304923631590136519156907, 10.03865159655229266833925810970
