| L(s) = 1 | + (0.888 − 0.458i)2-s + (−0.301 + 1.70i)3-s + (0.580 − 0.814i)4-s + (−0.115 − 0.475i)5-s + (0.513 + 1.65i)6-s + (1.23 + 3.57i)7-s + (0.142 − 0.989i)8-s + (−2.81 − 1.02i)9-s + (−0.320 − 0.370i)10-s + (0.226 + 4.76i)11-s + (1.21 + 1.23i)12-s + (0.731 − 2.11i)13-s + (2.74 + 2.61i)14-s + (0.846 − 0.0535i)15-s + (−0.327 − 0.945i)16-s + (0.843 − 1.84i)17-s + ⋯ |
| L(s) = 1 | + (0.628 − 0.324i)2-s + (−0.173 + 0.984i)3-s + (0.290 − 0.407i)4-s + (−0.0516 − 0.212i)5-s + (0.209 + 0.675i)6-s + (0.468 + 1.35i)7-s + (0.0503 − 0.349i)8-s + (−0.939 − 0.342i)9-s + (−0.101 − 0.117i)10-s + (0.0683 + 1.43i)11-s + (0.350 + 0.356i)12-s + (0.202 − 0.585i)13-s + (0.732 + 0.698i)14-s + (0.218 − 0.0138i)15-s + (−0.0817 − 0.236i)16-s + (0.204 − 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59672 + 0.918812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59672 + 0.918812i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.888 + 0.458i)T \) |
| 3 | \( 1 + (0.301 - 1.70i)T \) |
| 23 | \( 1 + (4.37 + 1.97i)T \) |
| good | 5 | \( 1 + (0.115 + 0.475i)T + (-4.44 + 2.29i)T^{2} \) |
| 7 | \( 1 + (-1.23 - 3.57i)T + (-5.50 + 4.32i)T^{2} \) |
| 11 | \( 1 + (-0.226 - 4.76i)T + (-10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (-0.731 + 2.11i)T + (-10.2 - 8.03i)T^{2} \) |
| 17 | \( 1 + (-0.843 + 1.84i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-3.03 - 6.65i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.962 - 1.35i)T + (-9.48 + 27.4i)T^{2} \) |
| 31 | \( 1 + (2.27 - 0.911i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (-4.11 + 1.20i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.653 + 2.69i)T + (-36.4 + 18.7i)T^{2} \) |
| 43 | \( 1 + (2.18 + 0.873i)T + (31.1 + 29.6i)T^{2} \) |
| 47 | \( 1 + (0.773 - 1.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.02 + 4.64i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.72 + 10.7i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (6.23 + 4.90i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (-0.483 + 10.1i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (12.1 + 7.81i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.05 - 13.2i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.19 + 0.615i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-1.81 + 7.48i)T + (-73.7 - 38.0i)T^{2} \) |
| 89 | \( 1 + (2.12 + 14.7i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-8.38 + 7.99i)T + (4.61 - 96.8i)T^{2} \) |
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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
−11.58536813167889096889794231775, −10.40088812728076943612048650048, −9.789259533561409539148777022553, −8.834426360158847846226622451899, −7.80737845008879692100054987800, −6.20068041022340794816053676410, −5.31987313253084588796589306266, −4.66378786773379452585849090244, −3.41256572143622985873974736515, −2.10283306514427778961551837627,
1.10685934053881239687394970445, 2.94092651836232904642688424212, 4.14066753395746545600857823163, 5.44269614561286324866547626968, 6.44430467473237283314019722273, 7.22139388142356617494531003842, 7.958223855078529331612817087650, 8.946995825006509316229592755846, 10.60112634922288799678620022580, 11.29171170249052843451137614627
