| L(s) = 1 | + (6.43 − 2.08i)2-s + (12.6 + 9.16i)3-s + (−14.7 + 10.7i)4-s + (−39.7 + 122. i)5-s + (100. + 32.5i)6-s + (156. + 215. i)7-s + (−327. + 450. i)8-s + (75.0 + 231. i)9-s + 869. i·10-s + (783. − 1.07e3i)11-s − 284.·12-s + (−9.21 + 2.99i)13-s + (1.45e3 + 1.05e3i)14-s + (−1.62e3 + 1.17e3i)15-s + (−801. + 2.46e3i)16-s + (3.72e3 + 1.21e3i)17-s + ⋯ |
| L(s) = 1 | + (0.803 − 0.261i)2-s + (0.467 + 0.339i)3-s + (−0.231 + 0.167i)4-s + (−0.317 + 0.977i)5-s + (0.464 + 0.150i)6-s + (0.456 + 0.628i)7-s + (−0.638 + 0.879i)8-s + (0.103 + 0.317i)9-s + 0.869i·10-s + (0.588 − 0.808i)11-s − 0.164·12-s + (−0.00419 + 0.00136i)13-s + (0.531 + 0.386i)14-s + (−0.480 + 0.348i)15-s + (−0.195 + 0.601i)16-s + (0.758 + 0.246i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.93880 + 1.43945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93880 + 1.43945i\) |
| \(L(4)\) |
|
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 | \( 1 + (-783. + 1.07e3i)T \) |
| good | 2 | \( 1 + (-6.43 + 2.08i)T + (51.7 - 37.6i)T^{2} \) |
| 5 | \( 1 + (39.7 - 122. i)T + (-1.26e4 - 9.18e3i)T^{2} \) |
| 7 | \( 1 + (-156. - 215. i)T + (-3.63e4 + 1.11e5i)T^{2} \) |
| 13 | \( 1 + (9.21 - 2.99i)T + (3.90e6 - 2.83e6i)T^{2} \) |
| 17 | \( 1 + (-3.72e3 - 1.21e3i)T + (1.95e7 + 1.41e7i)T^{2} \) |
| 19 | \( 1 + (-668. + 920. i)T + (-1.45e7 - 4.47e7i)T^{2} \) |
| 23 | \( 1 + 1.48e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (2.39e3 + 3.29e3i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 31 | \( 1 + (1.39e4 + 4.27e4i)T + (-7.18e8 + 5.21e8i)T^{2} \) |
| 37 | \( 1 + (-6.69e4 + 4.86e4i)T + (7.92e8 - 2.44e9i)T^{2} \) |
| 41 | \( 1 + (5.57e4 - 7.66e4i)T + (-1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + 1.26e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.07e5 - 7.78e4i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-8.74e4 - 2.69e5i)T + (-1.79e10 + 1.30e10i)T^{2} \) |
| 59 | \( 1 + (6.27e4 - 4.56e4i)T + (1.30e10 - 4.01e10i)T^{2} \) |
| 61 | \( 1 + (-3.19e5 - 1.03e5i)T + (4.16e10 + 3.02e10i)T^{2} \) |
| 67 | \( 1 + 5.70e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-1.58e5 + 4.88e5i)T + (-1.03e11 - 7.52e10i)T^{2} \) |
| 73 | \( 1 + (3.10e5 + 4.27e5i)T + (-4.67e10 + 1.43e11i)T^{2} \) |
| 79 | \( 1 + (3.06e5 - 9.95e4i)T + (1.96e11 - 1.42e11i)T^{2} \) |
| 83 | \( 1 + (-3.63e5 - 1.18e5i)T + (2.64e11 + 1.92e11i)T^{2} \) |
| 89 | \( 1 - 1.48e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (1.72e4 + 5.30e4i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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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
−15.05089939104176176640022696812, −14.55250768299024359519989301667, −13.45698094273592473007346628817, −11.96212400652841940632064592398, −11.00149115386044045756115202323, −9.183115082116882462872873976294, −7.87976080501100275261307333587, −5.78570797389882363084646840466, −3.99397083627675960095482638404, −2.76801717498812242170942855119,
1.07295073140091384994691490562, 3.92635764500730994448872743778, 5.11575672130878856022937962593, 7.03526298506261704864781303614, 8.568581410942409194531174102444, 9.886267389717123047044069878844, 12.01697733201192675322299987449, 12.89418879136978013342841159719, 14.03805051161942615421861503318, 14.82742625288384246398541115113
