L(s) = 1 | + (−4.27 + 2.46i)2-s + (8.16 − 14.1i)4-s + (7.65 − 13.2i)5-s + (6.32 + 17.4i)7-s + 41.0i·8-s + 75.5i·10-s + (36.9 − 21.3i)11-s + (−24.2 − 14.0i)13-s + (−69.9 − 58.7i)14-s + (−35.9 − 62.2i)16-s − 82.0·17-s − 113. i·19-s + (−125. − 216. i)20-s + (−105. + 182. i)22-s + (25.2 + 14.5i)23-s + ⋯ |
L(s) = 1 | + (−1.51 + 0.871i)2-s + (1.02 − 1.76i)4-s + (0.684 − 1.18i)5-s + (0.341 + 0.939i)7-s + 1.81i·8-s + 2.38i·10-s + (1.01 − 0.584i)11-s + (−0.517 − 0.299i)13-s + (−1.33 − 1.12i)14-s + (−0.562 − 0.973i)16-s − 1.17·17-s − 1.36i·19-s + (−1.39 − 2.42i)20-s + (−1.01 + 1.76i)22-s + (0.228 + 0.132i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.789741 - 0.270897i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.789741 - 0.270897i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-6.32 - 17.4i)T \) |
good | 2 | \( 1 + (4.27 - 2.46i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-7.65 + 13.2i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-36.9 + 21.3i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (24.2 + 14.0i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 82.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 113. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-25.2 - 14.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (16.5 - 9.56i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-97.0 - 56.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 69.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-242. + 419. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (194. + 337. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-53.4 - 92.5i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 207. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-97.8 + 169. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-446. + 257. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-221. + 383. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 640. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 528. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-20.7 - 35.8i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (554. + 960. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 121.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (748. - 432. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.79241618095413595011135414140, −10.73669184070102774048249619998, −9.247824336999200285740642943622, −9.123137318973382522401136711256, −8.331085199672194010499949314245, −6.93438407722834366310187631610, −5.91127017542577253451690468197, −4.91605113887828260009132529977, −2.04232355955255057886558625820, −0.64513771492367484492715866718,
1.42679335814904990960303104137, 2.56996781241155470397638534895, 4.10063547229906116993271554684, 6.50951333783169205134370122195, 7.22839525852954326248130461183, 8.344087155594092642816755750196, 9.746963585972337653715924248976, 9.981846865435069903167883961531, 11.07046085923179752688711557675, 11.61067118976384507230523892011