| L(s) = 1 | + (0.457 − 0.424i)2-s + (−22.4 + 3.38i)3-s + (−2.36 + 31.5i)4-s + (−1.17 + 2.98i)5-s + (−8.84 + 11.0i)6-s + (−27.2 − 126. i)7-s + (24.7 + 31.0i)8-s + (261. − 80.8i)9-s + (0.731 + 1.86i)10-s + (379. + 117. i)11-s + (−53.7 − 716. i)12-s + (−249. − 1.09e3i)13-s + (−66.2 − 46.4i)14-s + (16.2 − 71.1i)15-s + (−975. − 147. i)16-s + (833. − 568. i)17-s + ⋯ |
| L(s) = 1 | + (0.0808 − 0.0750i)2-s + (−1.44 + 0.217i)3-s + (−0.0738 + 0.985i)4-s + (−0.0209 + 0.0534i)5-s + (−0.100 + 0.125i)6-s + (−0.209 − 0.977i)7-s + (0.136 + 0.171i)8-s + (1.07 − 0.332i)9-s + (0.00231 + 0.00589i)10-s + (0.945 + 0.291i)11-s + (−0.107 − 1.43i)12-s + (−0.408 − 1.79i)13-s + (−0.0903 − 0.0633i)14-s + (0.0186 − 0.0816i)15-s + (−0.952 − 0.143i)16-s + (0.699 − 0.477i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.526174 - 0.412401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.526174 - 0.412401i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (27.2 + 126. i)T \) |
| good | 2 | \( 1 + (-0.457 + 0.424i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (22.4 - 3.38i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (1.17 - 2.98i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (-379. - 117. i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (249. + 1.09e3i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-833. + 568. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-303. - 524. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (3.14e3 + 2.14e3i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (-1.42e3 - 688. i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-516. + 894. i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (720. + 9.61e3i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (9.48e3 + 1.18e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (5.07e3 - 6.36e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (3.47e3 - 3.22e3i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (-1.92e3 + 2.57e4i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (1.61e4 + 4.11e4i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (2.35e3 + 3.13e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (8.98e3 - 1.55e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.89e4 + 1.39e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (2.85e3 + 2.64e3i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (-4.00e4 - 6.93e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.61e4 - 7.05e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-9.16e4 + 2.82e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 - 5.51e4T + 8.58e9T^{2} \) |
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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
−14.25975135133477358070820707552, −12.75676524477231459268846754162, −12.11332750035289878854069607582, −10.92658961505316232517303991480, −9.918388755586582642691915584975, −7.893984441943319039781037794756, −6.68398583228713150613984988846, −5.07660699258830877677651200573, −3.60768009556012022426522817696, −0.43305781706285710646487757532,
1.46137924777756407872689720362, 4.69003322182665458742794355453, 5.97421367106188598778988198448, 6.63658326709171301234062183001, 9.048556298379340437657529792196, 10.19610807638507974159807931398, 11.73502059350246158224297324810, 11.93806777264690139393307039194, 13.77639889116418645776031782442, 14.87052175497185836400738884401
