| L(s) = 1 | + (−0.431 + 1.34i)2-s + (−0.448 − 3.12i)3-s + (−1.62 − 1.16i)4-s + (−2.15 + 0.584i)5-s + (4.39 + 0.743i)6-s + (−0.713 − 0.618i)7-s + (2.26 − 1.68i)8-s + (−6.66 + 1.95i)9-s + (0.144 − 3.15i)10-s + (0.193 − 0.124i)11-s + (−2.90 + 5.60i)12-s + (−1.19 + 1.03i)13-s + (1.14 − 0.693i)14-s + (2.79 + 6.47i)15-s + (1.29 + 3.78i)16-s + (−0.523 + 1.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.305 + 0.952i)2-s + (−0.259 − 1.80i)3-s + (−0.813 − 0.581i)4-s + (−0.965 + 0.261i)5-s + (1.79 + 0.303i)6-s + (−0.269 − 0.233i)7-s + (0.802 − 0.597i)8-s + (−2.22 + 0.652i)9-s + (0.0457 − 0.998i)10-s + (0.0582 − 0.0374i)11-s + (−0.837 + 1.61i)12-s + (−0.331 + 0.287i)13-s + (0.304 − 0.185i)14-s + (0.721 + 1.67i)15-s + (0.323 + 0.946i)16-s + (−0.126 + 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0616465 + 0.135168i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0616465 + 0.135168i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.431 - 1.34i)T \) |
| 5 | \( 1 + (2.15 - 0.584i)T \) |
| 23 | \( 1 + (-4.74 - 0.709i)T \) |
| good | 3 | \( 1 + (0.448 + 3.12i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (0.713 + 0.618i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.193 + 0.124i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.19 - 1.03i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.523 - 1.14i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.95 - 4.28i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (1.83 - 4.00i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (7.75 + 1.11i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (8.70 - 2.55i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-9.64 - 2.83i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (9.31 - 1.33i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 + (7.74 - 8.93i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.50 + 2.16i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (9.64 + 1.38i)T + (58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.05 + 3.19i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-4.79 + 7.46i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.43 - 2.02i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.04 - 1.20i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.146 - 0.500i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (5.84 - 0.840i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (11.0 + 3.24i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.48792962727471815212083634150, −10.63221339461834132750256277043, −9.179916705005035743959618655121, −8.227738658498849271331558326960, −7.50601502649989152539607579884, −6.99720140162024270334636939814, −6.20253558946858841342886108639, −5.08416845631569479423770829131, −3.44142612723008375711309291519, −1.46747929233049335316832989546,
0.11055637447660226952824838961, 2.93946549569836546009543608259, 3.72864775549017350642421256267, 4.69442423838939569289669339406, 5.34706631552106318089457289357, 7.30011715968147842049882996572, 8.586138337264702994513592591328, 9.177609917733397763574382158056, 9.841857847577667244735259105280, 10.94052965553409054448590714454
