L(s) = 1 | − 2.18i·2-s − 9.62·3-s + 3.22·4-s − 10.6i·5-s + 21.0i·6-s − 26.7i·7-s − 24.5i·8-s + 65.6·9-s − 23.3·10-s + 11i·11-s − 31.0·12-s + (−34.6 + 31.5i)13-s − 58.3·14-s + 102. i·15-s − 27.7·16-s − 123.·17-s + ⋯ |
L(s) = 1 | − 0.772i·2-s − 1.85·3-s + 0.403·4-s − 0.956i·5-s + 1.43i·6-s − 1.44i·7-s − 1.08i·8-s + 2.43·9-s − 0.739·10-s + 0.301i·11-s − 0.746·12-s + (−0.740 + 0.672i)13-s − 1.11·14-s + 1.77i·15-s − 0.434·16-s − 1.76·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.219886 + 0.496738i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.219886 + 0.496738i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - 11iT \) |
| 13 | \( 1 + (34.6 - 31.5i)T \) |
good | 2 | \( 1 + 2.18iT - 8T^{2} \) |
| 3 | \( 1 + 9.62T + 27T^{2} \) |
| 5 | \( 1 + 10.6iT - 125T^{2} \) |
| 7 | \( 1 + 26.7iT - 343T^{2} \) |
| 17 | \( 1 + 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.83iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 184.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 90.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 98.3iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 313. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 60.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 601. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 128. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 102.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 186. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 736. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 814. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 171.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 15.5iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 878. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 750. iT - 9.12e5T^{2} \) |
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
−11.75422605213346003697788283971, −11.11215574088034308903564534500, −10.42227235676328145875166426329, −9.379168068825610360101713698500, −7.13061981896630780792683885477, −6.69832892206498975418656039350, −4.95188011284490709221797849581, −4.22956353831297430533787536785, −1.49941732756318684187410309771, −0.33537639182211852991229869482,
2.44382616022503851858901362942, 5.01376947669990166914508683599, 5.83492203258011305172588262748, 6.58739000378986512032772926609, 7.43222745882529604629894474737, 9.154314986808024232726617981635, 10.85611115140006037414827296838, 11.05930768972838564525010691009, 12.06099648880668481657787427080, 12.97466330091599098432156738448