L(s) = 1 | + (0.959 − 0.281i)2-s + (0.221 + 1.54i)3-s + (0.841 − 0.540i)4-s + (0.415 − 0.909i)5-s + (0.647 + 1.41i)6-s + (3.39 − 0.996i)7-s + (0.654 − 0.755i)8-s + (0.548 − 0.161i)9-s + (0.142 − 0.989i)10-s + (0.132 − 0.289i)11-s + (1.02 + 1.17i)12-s + (−2.45 − 2.83i)13-s + (2.97 − 1.91i)14-s + (1.49 + 0.438i)15-s + (0.415 − 0.909i)16-s + (−2.71 − 1.74i)17-s + ⋯ |
L(s) = 1 | + (0.678 − 0.199i)2-s + (0.128 + 0.890i)3-s + (0.420 − 0.270i)4-s + (0.185 − 0.406i)5-s + (0.264 + 0.578i)6-s + (1.28 − 0.376i)7-s + (0.231 − 0.267i)8-s + (0.182 − 0.0537i)9-s + (0.0450 − 0.313i)10-s + (0.0399 − 0.0874i)11-s + (0.294 + 0.339i)12-s + (−0.680 − 0.785i)13-s + (0.794 − 0.510i)14-s + (0.386 + 0.113i)15-s + (0.103 − 0.227i)16-s + (−0.657 − 0.422i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.71020 - 0.104954i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.71020 - 0.104954i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 67 | \( 1 + (-7.82 - 2.40i)T \) |
good | 3 | \( 1 + (-0.221 - 1.54i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-3.39 + 0.996i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.132 + 0.289i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (2.45 + 2.83i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.71 + 1.74i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.13 + 0.331i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (-0.542 - 3.77i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 + (5.82 - 6.71i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 + (4.03 + 2.59i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.89 - 1.22i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-0.283 - 1.97i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (9.96 - 6.40i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.72 - 3.14i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.997 - 2.18i)T + (-39.9 + 46.1i)T^{2} \) |
| 71 | \( 1 + (8.23 - 5.29i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.552 + 1.20i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (4.84 + 5.59i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.72 + 12.5i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (2.47 - 17.1i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−10.64437833056907715616574694689, −9.788195112142838297632006089434, −8.914999065913214680170718705711, −7.87086963776844901999705497546, −6.96904143265350074573134435852, −5.53287677777221556454197123038, −4.79743604896411978076372433757, −4.24847937637634851570104069220, −2.95941296536023095519874533614, −1.45696219198519267352188136873,
1.78466741075259650430786012357, 2.44994779769621676946157871492, 4.21579925527124804513558455904, 4.95484332803901857083411701710, 6.22316713606229957433051957533, 6.88635207645769678478128508445, 7.77844742447806258079888288948, 8.426057155017220783608673003585, 9.655220428425701341799297630302, 10.84455325789048915006906242329