L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 3·5-s + (−1 − 1.73i)7-s + 0.999·8-s + (1.5 − 2.59i)10-s + (3 − 5.19i)11-s + (−3.5 − 0.866i)13-s + 1.99·14-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + (−1 − 1.73i)19-s + (1.49 + 2.59i)20-s + (3 + 5.19i)22-s + (−3 + 5.19i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.34·5-s + (−0.377 − 0.654i)7-s + 0.353·8-s + (0.474 − 0.821i)10-s + (0.904 − 1.56i)11-s + (−0.970 − 0.240i)13-s + 0.534·14-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + (−0.229 − 0.397i)19-s + (0.335 + 0.580i)20-s + (0.639 + 1.10i)22-s + (−0.625 + 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.334186 - 0.329927i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.334186 - 0.329927i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (3.5 + 0.866i)T \) |
good | 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 13T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.63276270980214762991015696917, −11.17832741358054566050376267986, −9.836818777502747529574618586906, −8.853142777934576354247284924440, −7.82767285016940267538961198231, −7.13990079255358863722784981618, −5.97518159424314938829509064091, −4.44483197465409882758114154698, −3.38865056796063806419457874041, −0.42483959575600070168581499162,
2.18399610482479991675454113784, 3.81683390233727424146071713129, 4.67472244034330497157520803280, 6.61970265997946304499289246913, 7.57496549671765148961166062392, 8.610618857048113929815110205797, 9.545388777549186914031581038672, 10.48377386266024410993595002912, 11.72367671680586198058791197587, 12.24823812735498583034212044870