| L(s) = 1 | + (−6.15 − 12.7i)2-s + (4.85 + 7.11i)3-s + (−85.6 + 107. i)4-s + (−85.2 + 91.9i)5-s + (61.1 − 105. i)6-s + (94.7 − 54.6i)7-s + (1.01e3 + 231. i)8-s + (239. − 609. i)9-s + (1.70e3 + 524. i)10-s + (596. + 748. i)11-s + (−1.18e3 − 88.4i)12-s + (1.14e3 − 353. i)13-s + (−1.28e3 − 874. i)14-s + (−1.06e3 − 161. i)15-s + (−1.32e3 − 5.82e3i)16-s + (1.92e3 − 1.78e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.769 − 1.59i)2-s + (0.179 + 0.263i)3-s + (−1.33 + 1.67i)4-s + (−0.682 + 0.735i)5-s + (0.283 − 0.490i)6-s + (0.276 − 0.159i)7-s + (1.98 + 0.452i)8-s + (0.328 − 0.836i)9-s + (1.70 + 0.524i)10-s + (0.448 + 0.562i)11-s + (−0.682 − 0.0511i)12-s + (0.521 − 0.160i)13-s + (−0.467 − 0.318i)14-s + (−0.316 − 0.0477i)15-s + (−0.324 − 1.42i)16-s + (0.392 − 0.364i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.885048 - 0.597250i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.885048 - 0.597250i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.11e4 + 6.80e4i)T \) |
| good | 2 | \( 1 + (6.15 + 12.7i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-4.85 - 7.11i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (85.2 - 91.9i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-94.7 + 54.6i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-596. - 748. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-1.14e3 + 353. i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (-1.92e3 + 1.78e3i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-320. + 125. i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (-1.17e4 + 1.77e3i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (-5.36e3 + 7.86e3i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (3.27e3 - 4.37e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (-6.10e4 - 3.52e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-1.13e5 + 5.44e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-6.96e4 + 8.73e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (7.49e3 + 2.31e3i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (8.54e3 + 3.74e4i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (7.85e4 - 5.88e3i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (-1.66e5 - 4.24e5i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (1.71e4 - 1.13e5i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-1.64e5 - 5.31e5i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (1.94e5 + 3.36e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (5.44e5 - 3.71e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (3.54e5 + 5.19e5i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (6.36e5 + 7.98e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.39156386877316512382734398574, −12.76285550169579389239575287113, −11.76378848763619386714267560735, −10.86660075342912741296392460938, −9.799026151810442113224699851465, −8.692800220668578710579831223873, −7.17314426415610272900506923661, −4.11295384748733768724511566429, −3.02379394413498074938984509185, −1.03234007094761672142631734586,
0.945386564401041283521133519517, 4.59153814738194479835598019082, 6.05156999033368838820310295209, 7.61229160565626399347851342038, 8.296110424413694780055769995970, 9.365799377687999318893703218684, 11.13217644872507667102728752643, 12.92006747375118979766868339022, 14.13994524166068724497772885122, 15.20774406041412335165540846817
