| L(s) = 1 | + (0.309 − 0.951i)2-s + (0.978 + 0.207i)3-s + (−0.809 − 0.587i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (3.57 − 1.59i)7-s + (−0.809 + 0.587i)8-s + (0.913 + 0.406i)9-s + (0.978 − 0.207i)10-s + (−0.683 + 6.50i)11-s + (−0.669 − 0.743i)12-s + (0.240 − 0.267i)13-s + (−0.408 − 3.89i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.671 − 6.39i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.564 + 0.120i)3-s + (−0.404 − 0.293i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (1.35 − 0.601i)7-s + (−0.286 + 0.207i)8-s + (0.304 + 0.135i)9-s + (0.309 − 0.0657i)10-s + (−0.206 + 1.96i)11-s + (−0.193 − 0.214i)12-s + (0.0667 − 0.0741i)13-s + (−0.109 − 1.03i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.162 − 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.40013 - 0.572249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.40013 - 0.572249i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-4.97 - 2.50i)T \) |
| good | 7 | \( 1 + (-3.57 + 1.59i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.683 - 6.50i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.240 + 0.267i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.671 + 6.39i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-3.78 - 4.20i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.08 + 0.786i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.0381 - 0.117i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.144 + 0.251i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (9.06 - 1.92i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (4.56 + 5.07i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (0.671 + 2.06i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.70 + 1.64i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-4.86 - 1.03i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + (5.99 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.88 - 3.51i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-1.61 + 15.3i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (0.678 + 6.45i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (4.32 - 0.918i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (5.07 + 3.68i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-11.0 - 8.05i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.02293124499610369140629374458, −9.505858632121196073202140943727, −8.312355569821211048469520170609, −7.51466314174239206157261225985, −6.86108999662187419580453937945, −5.04663071004038797803003933157, −4.78086590824409549422924651294, −3.58225641215675806587297469259, −2.36901326351467650347504069951, −1.49535069604546569610209985093,
1.28158618085490548207833998856, 2.75745010284648526187861814438, 3.91213771195508048067200429280, 5.07094366493710844377571112978, 5.69767145275963736978853640449, 6.64608917511650485970855071680, 7.936970099112989599352430951782, 8.457580188277272468362945281439, 8.753509103468891922764827444657, 9.944057934984678322054161146565
