| L(s) = 1 | + (0.936 − 1.05i)2-s + (1.02 + 1.39i)3-s + (−0.246 − 1.98i)4-s + (1.49 + 1.66i)5-s + (2.43 + 0.215i)6-s − 4.44i·7-s + (−2.33 − 1.59i)8-s + (−0.886 + 2.86i)9-s + (3.16 − 0.0306i)10-s + (1.38 + 4.26i)11-s + (2.51 − 2.38i)12-s + (1.09 − 3.38i)13-s + (−4.71 − 4.16i)14-s + (−0.777 + 3.79i)15-s + (−3.87 + 0.979i)16-s + (−0.229 + 0.316i)17-s + ⋯ |
| L(s) = 1 | + (0.662 − 0.749i)2-s + (0.593 + 0.804i)3-s + (−0.123 − 0.992i)4-s + (0.669 + 0.743i)5-s + (0.996 + 0.0880i)6-s − 1.68i·7-s + (−0.825 − 0.564i)8-s + (−0.295 + 0.955i)9-s + (0.999 − 0.00969i)10-s + (0.417 + 1.28i)11-s + (0.725 − 0.688i)12-s + (0.304 − 0.938i)13-s + (−1.26 − 1.11i)14-s + (−0.200 + 0.979i)15-s + (−0.969 + 0.244i)16-s + (−0.0557 + 0.0767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.16783 - 0.598879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16783 - 0.598879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.936 + 1.05i)T \) |
| 3 | \( 1 + (-1.02 - 1.39i)T \) |
| 5 | \( 1 + (-1.49 - 1.66i)T \) |
| good | 7 | \( 1 + 4.44iT - 7T^{2} \) |
| 11 | \( 1 + (-1.38 - 4.26i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.09 + 3.38i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.229 - 0.316i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.18 - 1.63i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.927 - 2.85i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (4.65 + 6.40i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.713 - 0.981i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.145 - 0.447i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (7.88 + 2.56i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.44iT - 43T^{2} \) |
| 47 | \( 1 + (5.11 - 3.71i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.31 - 1.81i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.692 - 2.13i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.80 + 5.55i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-4.83 + 6.65i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (1.00 - 0.727i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.53 - 13.9i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.87 - 8.07i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.74 - 3.44i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-13.5 + 4.39i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-9.45 + 6.86i)T + (29.9 - 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.36023414052472287452543730286, −10.52383024569078340005902027174, −10.09304037505435791277772178153, −9.415954023525777514880216593901, −7.73111950329506128560058907439, −6.68683578100147255438194271768, −5.29966972963131345281072472123, −4.12350423280189764057787228455, −3.39436434433253160672219260935, −1.89777644891721726689409543203,
2.06093747166273946439166534010, 3.33132009683017380325301124407, 5.03392874195915057322776434309, 6.04247916004256531356197603698, 6.58888022268425633059753720922, 8.165324066259890312379106905136, 8.971710426939605301239534207777, 9.074097254166006616612471875833, 11.46236595567005066167733347896, 12.09615152898091712866973795281
