| L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (3.27 − 2.10i)5-s + (−0.654 + 0.755i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (0.553 + 3.84i)10-s + (2.21 + 4.85i)11-s + (−0.415 − 0.909i)12-s + (−0.760 − 5.29i)13-s + (−0.841 − 0.540i)14-s + (3.73 − 1.09i)15-s + (−0.142 + 0.989i)16-s + (−0.865 + 0.998i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (1.46 − 0.940i)5-s + (−0.267 + 0.308i)6-s + (−0.0537 + 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.175 + 1.21i)10-s + (0.667 + 1.46i)11-s + (−0.119 − 0.262i)12-s + (−0.210 − 1.46i)13-s + (−0.224 − 0.144i)14-s + (0.963 − 0.282i)15-s + (−0.0355 + 0.247i)16-s + (−0.209 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.03346 + 0.684767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03346 + 0.684767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (4.52 - 1.59i)T \) |
| good | 5 | \( 1 + (-3.27 + 2.10i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-2.21 - 4.85i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.760 + 5.29i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.865 - 0.998i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.77 - 4.35i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.33 + 3.84i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (2.44 - 0.718i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.15 - 2.66i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-4.00 + 2.57i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (8.92 + 2.62i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 8.51T + 47T^{2} \) |
| 53 | \( 1 + (-0.716 + 4.98i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (2.11 + 14.7i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-8.35 + 2.45i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (4.27 - 9.35i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (4.14 - 9.07i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-5.47 - 6.31i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.0994 + 0.691i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (7.26 + 4.66i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (5.61 + 1.64i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (15.6 - 10.0i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.887276903501165298935790866593, −9.393223023121785308082019563081, −8.442421255269602733861767971350, −7.80629686812242708643817265326, −6.66631851573621876816375670332, −5.67464401284202981760869709628, −5.17212288157573859370496945464, −4.02998084074056158700974025396, −2.37003094528881491210288552837, −1.38891732654941429292523915498,
1.32220430798355834712690423766, 2.43742979534312970590723253033, 3.19719979282098398447436242396, 4.34973400157902307467276775733, 5.80321731554964859797245261154, 6.62458304079127852869075940188, 7.29731714263572910638657839888, 8.647807423357194752227618898713, 9.276960231994978654601559057570, 9.777332520700718114377580278394
