| L(s) = 1 | + (0.121 + 0.210i)2-s + (32.0 + 18.5i)3-s + (31.9 − 55.3i)4-s + (−143. + 82.8i)5-s + 8.98i·6-s + (−197. − 280. i)7-s + 31.0·8-s + (321. + 556. i)9-s + (−34.8 − 20.1i)10-s + (86.5 − 149. i)11-s + (2.05e3 − 1.18e3i)12-s + 1.96e3i·13-s + (35.0 − 75.4i)14-s − 6.14e3·15-s + (−2.04e3 − 3.53e3i)16-s + (3.19e3 + 1.84e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0151 + 0.0262i)2-s + (1.18 + 0.685i)3-s + (0.499 − 0.865i)4-s + (−1.14 + 0.663i)5-s + 0.0416i·6-s + (−0.575 − 0.818i)7-s + 0.0606·8-s + (0.440 + 0.763i)9-s + (−0.0348 − 0.0201i)10-s + (0.0650 − 0.112i)11-s + (1.18 − 0.685i)12-s + 0.893i·13-s + (0.0127 − 0.0275i)14-s − 1.81·15-s + (−0.498 − 0.863i)16-s + (0.651 + 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.44012 + 0.156121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44012 + 0.156121i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (197. + 280. i)T \) |
| good | 2 | \( 1 + (-0.121 - 0.210i)T + (-32 + 55.4i)T^{2} \) |
| 3 | \( 1 + (-32.0 - 18.5i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (143. - 82.8i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-86.5 + 149. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 1.96e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-3.19e3 - 1.84e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-3.95e3 + 2.28e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-7.89e3 - 1.36e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.37e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.78e3 - 1.03e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (2.42e4 + 4.20e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 2.64e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.84e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (1.21e5 - 7.01e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.27e5 + 2.20e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (8.41e4 + 4.86e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.50e4 + 8.69e3i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-6.03e4 + 1.04e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 3.39e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-9.64e4 - 5.56e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-3.07e5 - 5.33e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 3.83e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (6.68e5 - 3.85e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 2.92e5iT - 8.32e11T^{2} \) |
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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
−20.92579368205458830251352454230, −19.61247076891366155112961049415, −19.21117065926860608948936331879, −16.14917196036740255821990508376, −15.10467496198957083154574766252, −14.05029234290685259194676965983, −11.17483326765565097196741844518, −9.610465856632806809670378816371, −7.29043247942431592939493297227, −3.58012065646708165250976690317,
3.10824522678675767063708019695, 7.57628376053108696110783887581, 8.656769052985805990756582332889, 12.04977263079773399468047393621, 12.97102011782892275808784118412, 15.15107196039987097751713589532, 16.40969323666429236651980339959, 18.65672919960831551765822913939, 19.89648662733088693126374806445, 20.70197342566805014342468607222
