| L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.913 + 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (2.70 − 3.00i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (−0.950 − 0.202i)11-s + (−0.104 + 0.994i)12-s + (−0.548 − 5.22i)13-s + (−3.95 + 0.841i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−1.60 + 0.341i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.527 + 0.234i)3-s + (0.154 + 0.475i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (1.02 − 1.13i)7-s + (0.109 − 0.336i)8-s + (0.223 + 0.247i)9-s + (0.288 − 0.128i)10-s + (−0.286 − 0.0609i)11-s + (−0.0301 + 0.287i)12-s + (−0.152 − 1.44i)13-s + (−1.05 + 0.224i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.389 + 0.0828i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10649 - 0.846563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10649 - 0.846563i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (5.08 - 2.27i)T \) |
| good | 7 | \( 1 + (-2.70 + 3.00i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (0.950 + 0.202i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.548 + 5.22i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (1.60 - 0.341i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.387 + 3.68i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.00 + 3.10i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.77 - 2.01i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-0.942 - 1.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.532 - 0.236i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.03 + 9.86i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-4.37 + 3.18i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.53 - 5.03i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-9.73 - 4.33i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 4.11T + 61T^{2} \) |
| 67 | \( 1 + (-0.581 + 1.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.80 + 7.55i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.919 - 0.195i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-3.39 + 0.721i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (4.21 - 1.87i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (1.97 + 6.08i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.30 - 16.3i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.23634565366165180819282159223, −8.934315176868750583140512851516, −8.287441779740354678212172747624, −7.50395423339964258676144386690, −6.96479026716853377962658849129, −5.31588512602468691392862127803, −4.34193519312065960202121331628, −3.35745540358371022367969401849, −2.32955652832000141763299411989, −0.78260114240516361922791715609,
1.56798581473074153971526880157, 2.40310976941465251869165287379, 4.09529720462220323262608081304, 5.08918554834856123126484369886, 5.98319297434309502084433589444, 7.08741372073958959684235644829, 7.905399327239956748824026943501, 8.560125496715688880686205086147, 9.164490656795670560277406297567, 9.877581449433459573322463247758
