| L(s) = 1 | + (7.29 + 5.81i)2-s + (−0.941 − 6.24i)3-s + (5.11 + 22.3i)4-s + (−69.5 − 101. i)5-s + (29.4 − 51.0i)6-s + (217. − 125. i)7-s + (166. − 344. i)8-s + (658. − 203. i)9-s + (86.0 − 1.14e3i)10-s + (−25.7 + 113. i)11-s + (135. − 53.0i)12-s + (−12.5 − 166. i)13-s + (2.31e3 + 348. i)14-s + (−571. + 530. i)15-s + (4.53e3 − 2.18e3i)16-s + (−5.41e3 − 3.69e3i)17-s + ⋯ |
| L(s) = 1 | + (0.911 + 0.726i)2-s + (−0.0348 − 0.231i)3-s + (0.0798 + 0.349i)4-s + (−0.556 − 0.815i)5-s + (0.136 − 0.236i)6-s + (0.633 − 0.366i)7-s + (0.324 − 0.673i)8-s + (0.903 − 0.278i)9-s + (0.0860 − 1.14i)10-s + (−0.0193 + 0.0849i)11-s + (0.0781 − 0.0306i)12-s + (−0.00569 − 0.0759i)13-s + (0.843 + 0.127i)14-s + (−0.169 + 0.157i)15-s + (1.10 − 0.533i)16-s + (−1.10 − 0.751i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.52020 - 0.754715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52020 - 0.754715i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.75e4 - 7.75e4i)T \) |
| good | 2 | \( 1 + (-7.29 - 5.81i)T + (14.2 + 62.3i)T^{2} \) |
| 3 | \( 1 + (0.941 + 6.24i)T + (-696. + 214. i)T^{2} \) |
| 5 | \( 1 + (69.5 + 101. i)T + (-5.70e3 + 1.45e4i)T^{2} \) |
| 7 | \( 1 + (-217. + 125. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (25.7 - 113. i)T + (-1.59e6 - 7.68e5i)T^{2} \) |
| 13 | \( 1 + (12.5 + 166. i)T + (-4.77e6 + 7.19e5i)T^{2} \) |
| 17 | \( 1 + (5.41e3 + 3.69e3i)T + (8.81e6 + 2.24e7i)T^{2} \) |
| 19 | \( 1 + (760. - 2.46e3i)T + (-3.88e7 - 2.65e7i)T^{2} \) |
| 23 | \( 1 + (-1.21e4 - 1.12e4i)T + (1.10e7 + 1.47e8i)T^{2} \) |
| 29 | \( 1 + (2.23e3 - 1.48e4i)T + (-5.68e8 - 1.75e8i)T^{2} \) |
| 31 | \( 1 + (1.20e4 + 3.06e4i)T + (-6.50e8 + 6.03e8i)T^{2} \) |
| 37 | \( 1 + (-2.20e4 - 1.27e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (2.49e4 - 3.12e4i)T + (-1.05e9 - 4.63e9i)T^{2} \) |
| 47 | \( 1 + (-3.73e4 - 1.63e5i)T + (-9.71e9 + 4.67e9i)T^{2} \) |
| 53 | \( 1 + (-1.60e4 + 2.13e5i)T + (-2.19e10 - 3.30e9i)T^{2} \) |
| 59 | \( 1 + (2.04e4 - 9.82e3i)T + (2.62e10 - 3.29e10i)T^{2} \) |
| 61 | \( 1 + (-3.13e5 - 1.23e5i)T + (3.77e10 + 3.50e10i)T^{2} \) |
| 67 | \( 1 + (-3.06e5 - 9.45e4i)T + (7.47e10 + 5.09e10i)T^{2} \) |
| 71 | \( 1 + (-5.43e4 - 5.85e4i)T + (-9.57e9 + 1.27e11i)T^{2} \) |
| 73 | \( 1 + (-2.60e5 + 1.95e4i)T + (1.49e11 - 2.25e10i)T^{2} \) |
| 79 | \( 1 + (-1.84e5 - 3.18e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-4.94e4 + 7.44e3i)T + (3.12e11 - 9.63e10i)T^{2} \) |
| 89 | \( 1 + (2.56e4 + 1.70e5i)T + (-4.74e11 + 1.46e11i)T^{2} \) |
| 97 | \( 1 + (2.76e5 - 1.21e6i)T + (-7.50e11 - 3.61e11i)T^{2} \) |
| 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
−14.67051421283801384680157103865, −13.36836504832373533195156943541, −12.69301943892981623563089494102, −11.27967556673669550811051659643, −9.546133641816599770775593386071, −7.85283238959594021942239731309, −6.76655763145114681652032282975, −5.02977425316372203255220484176, −4.13165550704671509433680055684, −1.05916631911766088880029253709,
2.17619442939172228880399287406, 3.78505535979710738328984516322, 4.95504020763963275928528493938, 7.01945932271300219722171663616, 8.536129216589432260365402593431, 10.60530437987118840256463633465, 11.21015545476460741016556871920, 12.44161050644705628452821267627, 13.47973858938165997494032249512, 14.76465076824614866985582782032
