| L(s) = 1 | + (4.51 − 1.46i)2-s + (−12.6 − 9.16i)3-s + (−33.5 + 24.3i)4-s + (6.35 − 19.5i)5-s + (−70.3 − 22.8i)6-s + (355. + 489. i)7-s + (−294. + 404. i)8-s + (75.0 + 231. i)9-s − 97.5i·10-s + (−598. + 1.18e3i)11-s + 646.·12-s + (654. − 212. i)13-s + (2.32e3 + 1.68e3i)14-s + (−259. + 188. i)15-s + (86.4 − 266. i)16-s + (−3.27e3 − 1.06e3i)17-s + ⋯ |
| L(s) = 1 | + (0.564 − 0.183i)2-s + (−0.467 − 0.339i)3-s + (−0.524 + 0.380i)4-s + (0.0508 − 0.156i)5-s + (−0.325 − 0.105i)6-s + (1.03 + 1.42i)7-s + (−0.574 + 0.790i)8-s + (0.103 + 0.317i)9-s − 0.0975i·10-s + (−0.449 + 0.893i)11-s + 0.374·12-s + (0.297 − 0.0967i)13-s + (0.847 + 0.615i)14-s + (−0.0768 + 0.0558i)15-s + (0.0211 − 0.0649i)16-s + (−0.666 − 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.05635 + 0.926872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05635 + 0.926872i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (12.6 + 9.16i)T \) |
| 11 | \( 1 + (598. - 1.18e3i)T \) |
| good | 2 | \( 1 + (-4.51 + 1.46i)T + (51.7 - 37.6i)T^{2} \) |
| 5 | \( 1 + (-6.35 + 19.5i)T + (-1.26e4 - 9.18e3i)T^{2} \) |
| 7 | \( 1 + (-355. - 489. i)T + (-3.63e4 + 1.11e5i)T^{2} \) |
| 13 | \( 1 + (-654. + 212. i)T + (3.90e6 - 2.83e6i)T^{2} \) |
| 17 | \( 1 + (3.27e3 + 1.06e3i)T + (1.95e7 + 1.41e7i)T^{2} \) |
| 19 | \( 1 + (4.16e3 - 5.72e3i)T + (-1.45e7 - 4.47e7i)T^{2} \) |
| 23 | \( 1 - 864.T + 1.48e8T^{2} \) |
| 29 | \( 1 + (8.64e3 + 1.18e4i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 31 | \( 1 + (5.05e3 + 1.55e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (5.77e3 - 4.19e3i)T + (7.92e8 - 2.44e9i)T^{2} \) |
| 41 | \( 1 + (-6.52e4 + 8.98e4i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 - 1.34e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.92e4 - 1.39e4i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-8.12e4 - 2.50e5i)T + (-1.79e10 + 1.30e10i)T^{2} \) |
| 59 | \( 1 + (-2.41e5 + 1.75e5i)T + (1.30e10 - 4.01e10i)T^{2} \) |
| 61 | \( 1 + (2.52e5 + 8.21e4i)T + (4.16e10 + 3.02e10i)T^{2} \) |
| 67 | \( 1 - 4.67e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-2.17e4 + 6.70e4i)T + (-1.03e11 - 7.52e10i)T^{2} \) |
| 73 | \( 1 + (5.85e4 + 8.05e4i)T + (-4.67e10 + 1.43e11i)T^{2} \) |
| 79 | \( 1 + (-4.95e5 + 1.61e5i)T + (1.96e11 - 1.42e11i)T^{2} \) |
| 83 | \( 1 + (4.65e5 + 1.51e5i)T + (2.64e11 + 1.92e11i)T^{2} \) |
| 89 | \( 1 - 3.28e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (2.55e5 + 7.85e5i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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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
−15.43904762417788959479350176533, −14.45227017970994735097621177971, −12.96007719636695520007418272447, −12.24656311690687525236266721111, −11.16015460036342287581878388805, −9.090961171990713285241978936333, −7.917049980513582027592085164922, −5.71597301796376100444133005013, −4.62457365673913842903511503741, −2.22217598979834216001592610331,
0.68102773889581617369886315149, 4.00782725676292980128444240849, 5.13064977076128059472562872795, 6.75413710438024508700672679436, 8.632401900874710253468526472063, 10.42405369128282359384756587924, 11.13422972514538039743733377467, 13.07834704181133625585481089103, 13.96620367816665136739857829140, 14.93545372707587000242431869303
