| L(s) = 1 | + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.515 + 3.58i)5-s + (0.841 + 0.540i)6-s + (0.415 − 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (1.50 + 3.29i)10-s + (2.78 + 0.817i)11-s + (0.959 + 0.281i)12-s + (0.667 + 1.46i)13-s + (0.142 − 0.989i)14-s + (−2.37 + 2.73i)15-s + (0.415 − 0.909i)16-s + (−1.52 − 0.981i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (0.230 + 1.60i)5-s + (0.343 + 0.220i)6-s + (0.157 − 0.343i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.476 + 1.04i)10-s + (0.839 + 0.246i)11-s + (0.276 + 0.0813i)12-s + (0.185 + 0.405i)13-s + (0.0380 − 0.264i)14-s + (−0.612 + 0.707i)15-s + (0.103 − 0.227i)16-s + (−0.370 − 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.47863 + 1.49592i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.47863 + 1.49592i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (1.19 + 4.64i)T \) |
| good | 5 | \( 1 + (-0.515 - 3.58i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-2.78 - 0.817i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.667 - 1.46i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (1.52 + 0.981i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (2.70 - 1.73i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.64 - 1.70i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (0.580 - 0.669i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.163 + 1.13i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.03 - 7.23i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.21 + 2.55i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + (-2.10 + 4.61i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (2.54 + 5.57i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (5.84 - 6.74i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.26 + 0.664i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (1.22 - 0.358i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-5.25 + 3.37i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.96 + 8.68i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.755 + 5.25i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (1.16 + 1.34i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.93 - 13.4i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.39670594528617778165778064801, −9.582408894238209517323860649601, −8.548228551465732478657071374107, −7.35099861561014014170731248645, −6.66939511218810140426286029751, −6.02511454362085415780468392228, −4.58147655177284552835597183949, −3.85269466128500921146309402598, −2.91037369630469579296355040362, −1.94960964301331895391551115236,
1.13002001175567518612654846150, 2.26964545215300292591592874965, 3.74807849192320329460901832771, 4.57898758900903909235108135269, 5.56212183675244677762682180353, 6.21803032450018229493873370314, 7.36475921099722755656360963388, 8.346623923468655573058035917258, 8.835426824752507789925141035777, 9.567068322743986697782307646912
