| L(s) = 1 | + (−0.358 − 0.933i)2-s + (−0.907 − 1.39i)3-s + (−0.743 + 0.669i)4-s + (1.00 + 1.99i)5-s + (−0.978 + 1.34i)6-s + (2.13 + 1.56i)7-s + (0.891 + 0.453i)8-s + (0.0921 − 0.207i)9-s + (1.50 − 1.65i)10-s + (−3.26 + 0.604i)11-s + (1.60 + 0.431i)12-s + (−2.67 − 0.424i)13-s + (0.697 − 2.55i)14-s + (1.87 − 3.21i)15-s + (0.104 − 0.994i)16-s + (−2.59 + 6.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 − 0.660i)2-s + (−0.523 − 0.806i)3-s + (−0.371 + 0.334i)4-s + (0.450 + 0.892i)5-s + (−0.399 + 0.550i)6-s + (0.806 + 0.591i)7-s + (0.315 + 0.160i)8-s + (0.0307 − 0.0690i)9-s + (0.475 − 0.523i)10-s + (−0.983 + 0.182i)11-s + (0.464 + 0.124i)12-s + (−0.742 − 0.117i)13-s + (0.186 − 0.682i)14-s + (0.483 − 0.830i)15-s + (0.0261 − 0.248i)16-s + (−0.630 + 1.64i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.614803 + 0.352237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.614803 + 0.352237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.358 + 0.933i)T \) |
| 5 | \( 1 + (-1.00 - 1.99i)T \) |
| 7 | \( 1 + (-2.13 - 1.56i)T \) |
| 11 | \( 1 + (3.26 - 0.604i)T \) |
| good | 3 | \( 1 + (0.907 + 1.39i)T + (-1.22 + 2.74i)T^{2} \) |
| 13 | \( 1 + (2.67 + 0.424i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.59 - 6.76i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (3.60 - 4.00i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.670 - 0.179i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-4.04 - 1.31i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.35 - 0.457i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (5.55 + 3.60i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-6.92 + 2.25i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (5.74 - 5.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.266 + 5.09i)T + (-46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (-1.11 - 0.900i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-0.117 - 0.130i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-12.0 - 1.26i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.98 + 0.800i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-6.84 - 4.97i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.808 - 15.4i)T + (-72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (-3.52 + 7.91i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (2.65 + 16.7i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.366 - 0.635i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.14 - 13.5i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−10.59280433353094213825348767132, −9.914785260233059811275456232600, −8.689054911226229586707358384594, −7.908982735002906060994633081301, −7.02317566646648082640643293424, −6.07402934706463183668811666314, −5.25205446966115107744676551494, −3.85586857277245801939387553723, −2.38456537401223611595522657616, −1.74523019127559896546727057394,
0.39790393632610384767673438000, 2.26990483705999659971481033016, 4.34403956327952118338534997746, 4.99355982789795432411083853271, 5.28473617232591511609908381573, 6.74029048637401304174374455342, 7.63382476379406691865478876207, 8.493038498758448651589961896681, 9.368562171018673693794122169528, 10.08039758500478942241600609815
