| L(s) = 1 | + (4.47 − 9.29i)2-s + (30.6 + 2.29i)3-s + (−26.4 − 33.1i)4-s + (−67.0 + 217. i)5-s + (158. − 274. i)6-s + (521. − 301. i)7-s + (217. − 49.5i)8-s + (214. + 32.2i)9-s + (1.72e3 + 1.59e3i)10-s + (1.09e3 − 1.36e3i)11-s + (−734. − 1.07e3i)12-s + (−2.01e3 + 1.87e3i)13-s + (−464. − 6.19e3i)14-s + (−2.55e3 + 6.51e3i)15-s + (1.11e3 − 4.88e3i)16-s + (−1.26e3 + 390. i)17-s + ⋯ |
| L(s) = 1 | + (0.559 − 1.16i)2-s + (1.13 + 0.0851i)3-s + (−0.413 − 0.518i)4-s + (−0.536 + 1.73i)5-s + (0.734 − 1.27i)6-s + (1.52 − 0.878i)7-s + (0.424 − 0.0967i)8-s + (0.293 + 0.0442i)9-s + (1.72 + 1.59i)10-s + (0.819 − 1.02i)11-s + (−0.425 − 0.623i)12-s + (−0.918 + 0.851i)13-s + (−0.169 − 2.25i)14-s + (−0.757 + 1.93i)15-s + (0.272 − 1.19i)16-s + (−0.257 + 0.0795i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.737i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.674 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.16630 - 1.39485i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.16630 - 1.39485i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.24e4 - 6.72e4i)T \) |
| good | 2 | \( 1 + (-4.47 + 9.29i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-30.6 - 2.29i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (67.0 - 217. i)T + (-1.29e4 - 8.80e3i)T^{2} \) |
| 7 | \( 1 + (-521. + 301. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.09e3 + 1.36e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (2.01e3 - 1.87e3i)T + (3.60e5 - 4.81e6i)T^{2} \) |
| 17 | \( 1 + (1.26e3 - 390. i)T + (1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (-897. - 5.95e3i)T + (-4.49e7 + 1.38e7i)T^{2} \) |
| 23 | \( 1 + (478. + 1.21e3i)T + (-1.08e8 + 1.00e8i)T^{2} \) |
| 29 | \( 1 + (-8.93e3 + 669. i)T + (5.88e8 - 8.86e7i)T^{2} \) |
| 31 | \( 1 + (6.18e3 - 4.21e3i)T + (3.24e8 - 8.26e8i)T^{2} \) |
| 37 | \( 1 + (7.27e3 + 4.19e3i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (3.37e4 + 1.62e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (7.47e4 + 9.37e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-1.45e4 - 1.34e4i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-4.43e4 + 1.94e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.44e5 - 2.11e5i)T + (-1.88e10 - 4.79e10i)T^{2} \) |
| 67 | \( 1 + (3.09e5 - 4.66e4i)T + (8.64e10 - 2.66e10i)T^{2} \) |
| 71 | \( 1 + (-3.87e5 - 1.51e5i)T + (9.39e10 + 8.71e10i)T^{2} \) |
| 73 | \( 1 + (-2.55e5 - 2.75e5i)T + (-1.13e10 + 1.50e11i)T^{2} \) |
| 79 | \( 1 + (1.13e5 + 1.97e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (2.66e4 - 3.55e5i)T + (-3.23e11 - 4.87e10i)T^{2} \) |
| 89 | \( 1 + (1.30e6 + 9.79e4i)T + (4.91e11 + 7.40e10i)T^{2} \) |
| 97 | \( 1 + (-1.56e5 + 1.95e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−14.16907651002602257537403047843, −13.94664235776849094813671376522, −11.69438491082626418409889145402, −11.21313682141483293741882132300, −10.11471158817498103040698145913, −8.172247073312535934200718761427, −7.10757225683045292460589897333, −4.16173920301171682489218444579, −3.26063721563740123691542447867, −1.92037573220870871041822913027,
1.77404566312659051456762755550, 4.54719830605325386345126396295, 5.21340947048102901831287482627, 7.56645441053553462733195612187, 8.309017415356166102514737549749, 9.174912654247534979938430909506, 11.75293525810941977447473471501, 12.80387060663831161153353602522, 14.04719189664183681141861786303, 15.09946139445848427285826379947
