L(s) = 1 | + (1.17 + 2.57i)2-s + (7.57 + 3.13i)3-s + (−5.22 + 6.05i)4-s + (−8.03 − 19.4i)5-s + (0.844 + 23.1i)6-s + (−1.85 + 1.85i)7-s + (−21.7 − 6.32i)8-s + (28.3 + 28.3i)9-s + (40.4 − 43.5i)10-s + (8.63 − 3.57i)11-s + (−58.5 + 29.4i)12-s + (−11.5 + 27.9i)13-s + (−6.93 − 2.58i)14-s − 172. i·15-s + (−9.29 − 63.3i)16-s − 7.99i·17-s + ⋯ |
L(s) = 1 | + (0.416 + 0.909i)2-s + (1.45 + 0.603i)3-s + (−0.653 + 0.756i)4-s + (−0.718 − 1.73i)5-s + (0.0574 + 1.57i)6-s + (−0.0999 + 0.0999i)7-s + (−0.960 − 0.279i)8-s + (1.05 + 1.05i)9-s + (1.27 − 1.37i)10-s + (0.236 − 0.0979i)11-s + (−1.40 + 0.707i)12-s + (−0.247 + 0.597i)13-s + (−0.132 − 0.0492i)14-s − 2.96i·15-s + (−0.145 − 0.989i)16-s − 0.114i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.52028 + 1.00043i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52028 + 1.00043i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.17 - 2.57i)T \) |
good | 3 | \( 1 + (-7.57 - 3.13i)T + (19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (8.03 + 19.4i)T + (-88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (1.85 - 1.85i)T - 343iT^{2} \) |
| 11 | \( 1 + (-8.63 + 3.57i)T + (941. - 941. i)T^{2} \) |
| 13 | \( 1 + (11.5 - 27.9i)T + (-1.55e3 - 1.55e3i)T^{2} \) |
| 17 | \( 1 + 7.99iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-5.76 + 13.9i)T + (-4.85e3 - 4.85e3i)T^{2} \) |
| 23 | \( 1 + (-60.2 - 60.2i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (-167. - 69.3i)T + (1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 + 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-0.431 - 1.04i)T + (-3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (275. + 275. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (-257. + 106. i)T + (5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 - 51.3iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (2.98 - 1.23i)T + (1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (-101. - 244. i)T + (-1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (270. + 111. i)T + (1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 + (-778. - 322. i)T + (2.12e5 + 2.12e5i)T^{2} \) |
| 71 | \( 1 + (-484. + 484. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (212. + 212. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 593. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-320. + 773. i)T + (-4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-435. + 435. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + 570.T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.11794100285676904627588744963, −15.49200328614128213330487761387, −14.28515467465189308963249721335, −13.23337121484621211728903596418, −12.12462321698778305725213156563, −9.237682105252411363914649952159, −8.783915345291148181353536776024, −7.59504536161378391696920565693, −4.94665738401804836568429794347, −3.77457883727207504260897840162,
2.58661485736058550420121867050, 3.61934487115080960383027731883, 6.81748616691275363812883213561, 8.191980276840900434330786687729, 9.889528498121141447559410477642, 11.11340150242056307482098554184, 12.53897219943770264230987195492, 13.83257193459346695911432441726, 14.64018123307421723179504876479, 15.25371219377940858655527268169