| 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
−9.777332520700718114377580278394, −9.276960231994978654601559057570, −8.647807423357194752227618898713, −7.29731714263572910638657839888, −6.62458304079127852869075940188, −5.80321731554964859797245261154, −4.34973400157902307467276775733, −3.19719979282098398447436242396, −2.43742979534312970590723253033, −1.32220430798355834712690423766,
1.38891732654941429292523915498, 2.37003094528881491210288552837, 4.02998084074056158700974025396, 5.17212288157573859370496945464, 5.67464401284202981760869709628, 6.66631851573621876816375670332, 7.80629686812242708643817265326, 8.442421255269602733861767971350, 9.393223023121785308082019563081, 9.887276903501165298935790866593
