L(s) = 1 | + (−0.866 − 1.5i)2-s + (−1 − 1.73i)3-s + (−0.5 + 0.866i)4-s − 1.73·5-s + (−1.73 + 3i)6-s − 1.73·8-s + (−0.499 + 0.866i)9-s + (1.49 + 2.59i)10-s + 2·12-s + (1.73 + 2.99i)15-s + (2.49 + 4.33i)16-s + (−1.5 + 2.59i)17-s + 1.73·18-s + (1.73 − 3i)19-s + (0.866 − 1.50i)20-s + ⋯ |
L(s) = 1 | + (−0.612 − 1.06i)2-s + (−0.577 − 0.999i)3-s + (−0.250 + 0.433i)4-s − 0.774·5-s + (−0.707 + 1.22i)6-s − 0.612·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + 0.577·12-s + (0.447 + 0.774i)15-s + (0.624 + 1.08i)16-s + (−0.363 + 0.630i)17-s + 0.408·18-s + (0.397 − 0.688i)19-s + (0.193 − 0.335i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.165213 + 0.368281i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.165213 + 0.368281i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 1.5i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.73 + 3i)T + (-9.5 - 16.4i)T^{2} \) |
| 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 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + (4.33 + 7.5i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.59 + 4.5i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (-3.46 + 6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 - 3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.73 - 3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + (-3.46 - 6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.46 - 6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.11720908568289323627167364369, −11.26444772344684466916008131603, −10.46892661763544810914590236874, −9.199571952202992873146175998705, −8.108558979652241302615090395837, −6.96304453987406784930278395185, −5.86009954120082912247415771514, −3.91237810568063505552473766069, −2.16476351842961474470588166149, −0.48155172633364216074792975011,
3.55982381203645878803030763207, 4.92344190111112481042795812599, 6.03641800405607795597681235072, 7.34049854202209080245702459206, 8.138632461951013330312426057722, 9.349665201872342313588955562620, 10.16326168720162575464679537427, 11.44949318530580681882544735116, 12.02330629613270342084878420756, 13.63354665267225313901414865356