| L(s) = 1 | + (−0.951 + 0.309i)2-s + (1.48 + 0.890i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (−1.68 − 0.387i)6-s + (0.0245 + 0.234i)7-s + (−0.587 + 0.809i)8-s + (1.41 + 2.64i)9-s + (−0.669 + 0.743i)10-s + (1.45 + 0.646i)11-s + (1.72 − 0.153i)12-s + (−1.21 − 5.69i)13-s + (−0.0957 − 0.214i)14-s + (1.73 + 0.0280i)15-s + (0.309 − 0.951i)16-s + (3.90 − 1.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.857 + 0.513i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (−0.689 − 0.158i)6-s + (0.00929 + 0.0884i)7-s + (−0.207 + 0.286i)8-s + (0.471 + 0.881i)9-s + (−0.211 + 0.235i)10-s + (0.437 + 0.194i)11-s + (0.498 − 0.0442i)12-s + (−0.335 − 1.58i)13-s + (−0.0255 − 0.0574i)14-s + (0.447 + 0.00723i)15-s + (0.0772 − 0.237i)16-s + (0.947 − 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70984 + 0.490465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70984 + 0.490465i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-1.48 - 0.890i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (5.48 - 0.974i)T \) |
| good | 7 | \( 1 + (-0.0245 - 0.234i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-1.45 - 0.646i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (1.21 + 5.69i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.90 + 1.73i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.572 - 0.121i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-7.46 - 5.42i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.549 - 1.69i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.834 - 0.481i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.53 + 4.98i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.201 + 0.945i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-7.81 - 2.53i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.193 - 1.84i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-2.88 + 2.59i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 5.54iT - 61T^{2} \) |
| 67 | \( 1 + (-2.84 - 4.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (13.9 + 1.46i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (6.83 - 15.3i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (5.16 + 11.5i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (10.2 - 11.3i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-14.8 + 10.7i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.67 + 4.11i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.06121196641977958784781466804, −9.177058173844486116571460060135, −8.719961918566082223255721654335, −7.60954501247114948886026543604, −7.21151382498408626578073971958, −5.61407990048654343372266824460, −5.10152418303174099160212193821, −3.55886122404360436715982929943, −2.69270957276760252009868047747, −1.26437164847662104350173570196,
1.23677987212791266896046915166, 2.27081857039759032477160525802, 3.30825090857292623803645903137, 4.41057679374750723414440292206, 6.01022378297902423720463743160, 6.92382440918781238943983435020, 7.39644574637828234546742000680, 8.593050811332878972036165308377, 9.052384743734414433796617243118, 9.785131941854045528376849376247
