| L(s) = 1 | + (−1.80 − 3.12i)2-s + (25.4 − 44.1i)4-s + (−71.9 + 41.5i)5-s + (77.0 + 334. i)7-s − 414.·8-s + (259. + 149. i)10-s + (−221. + 383. i)11-s + 696. i·13-s + (904. − 843. i)14-s + (−883. − 1.53e3i)16-s + (5.44e3 + 3.14e3i)17-s + (−2.32e3 + 1.34e3i)19-s + 4.23e3i·20-s + 1.59e3·22-s + (7.88e3 + 1.36e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.225 − 0.390i)2-s + (0.398 − 0.690i)4-s + (−0.575 + 0.332i)5-s + (0.224 + 0.974i)7-s − 0.810·8-s + (0.259 + 0.149i)10-s + (−0.166 + 0.287i)11-s + 0.317i·13-s + (0.329 − 0.307i)14-s + (−0.215 − 0.373i)16-s + (1.10 + 0.640i)17-s + (−0.339 + 0.195i)19-s + 0.529i·20-s + 0.149·22-s + (0.648 + 1.12i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.14406 + 0.536535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14406 + 0.536535i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-77.0 - 334. i)T \) |
| good | 2 | \( 1 + (1.80 + 3.12i)T + (-32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (71.9 - 41.5i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (221. - 383. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 696. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-5.44e3 - 3.14e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (2.32e3 - 1.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-7.88e3 - 1.36e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 2.32e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-4.11e4 - 2.37e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-5.07e3 - 8.79e3i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 3.81e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.51e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-4.35e4 + 2.51e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (9.97e4 - 1.72e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (3.35e5 + 1.93e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.02e4 - 5.89e3i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.92e5 + 3.32e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.56e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-3.25e5 - 1.87e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.65e4 + 2.87e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 9.84e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-2.27e5 + 1.31e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 5.75e5iT - 8.32e11T^{2} \) |
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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
−14.09249239652814510402405192303, −12.32808598898879611441673056037, −11.63707097021377280730975487857, −10.53019664380228145405595180579, −9.418441524504731235210251402136, −8.052177812396849768702908629192, −6.51219198544927868489810991270, −5.20887882079749377586893256440, −3.10630892935579483322615554115, −1.53681210692477839869878743345,
0.57306509378343003738053809781, 3.06375428261589047290131672149, 4.55371846163410223625969655702, 6.51674809303162810195915216908, 7.72855243609891284121386525183, 8.425968827045963100631035220949, 10.14010484617800206597848272222, 11.43598740293679838167358961334, 12.36571789563120382643569206737, 13.54546278572127146436578064480
