L(s) = 1 | + (0.392 − 0.680i)2-s + (1.02 + 1.39i)3-s + (0.691 + 1.19i)4-s + (−0.5 − 0.866i)5-s + (1.35 − 0.153i)6-s + (−0.5 + 0.866i)7-s + 2.65·8-s + (−0.880 + 2.86i)9-s − 0.785·10-s + (1.09 − 1.89i)11-s + (−0.956 + 2.19i)12-s + (0.415 + 0.719i)13-s + (0.392 + 0.680i)14-s + (0.691 − 1.58i)15-s + (−0.340 + 0.589i)16-s + 0.994·17-s + ⋯ |
L(s) = 1 | + (0.277 − 0.480i)2-s + (0.594 + 0.804i)3-s + (0.345 + 0.599i)4-s + (−0.223 − 0.387i)5-s + (0.551 − 0.0625i)6-s + (−0.188 + 0.327i)7-s + 0.939·8-s + (−0.293 + 0.955i)9-s − 0.248·10-s + (0.330 − 0.572i)11-s + (−0.276 + 0.634i)12-s + (0.115 + 0.199i)13-s + (0.104 + 0.181i)14-s + (0.178 − 0.410i)15-s + (−0.0850 + 0.147i)16-s + 0.241·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82960 + 0.540665i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82960 + 0.540665i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.02 - 1.39i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.392 + 0.680i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.09 + 1.89i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 0.719i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.994T + 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.02 + 5.24i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.857 + 1.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 + (5.67 + 9.83i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.85 + 8.41i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.09 + 1.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2.95T + 53T^{2} \) |
| 59 | \( 1 + (2.75 + 4.76i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.66 + 6.34i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.60 - 6.24i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + (3.91 - 6.77i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.41 - 12.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + (5.55 - 9.61i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.72480940507407183760034351391, −10.91103836348147737180229005721, −9.974912184109263569125694846965, −8.810599811031527520211643301151, −8.260394764846553946199262088710, −7.06154947343856845297986050035, −5.52882844206350050434309329092, −4.23025479563775397326988163195, −3.47546807124902971248779218866, −2.23043764810094226181264876317,
1.47684946099260115608005256086, 2.99864511407913339190359926809, 4.49076009788408072394229732319, 5.97795557127560996124585469444, 6.82368129403318477429435230168, 7.42280884666841811775090703747, 8.513277495519568955929706864866, 9.739080969345307897584287912305, 10.62064493186161278094050756733, 11.66130025623705467542815003363