| L(s) = 1 | + (−1.34 + 3.43i)2-s + (−1.28 − 0.0960i)3-s + (36.9 + 34.2i)4-s + (103. − 151. i)5-s + (2.05 − 4.27i)6-s + (−113. − 323. i)7-s + (−380. + 183. i)8-s + (−719. − 108. i)9-s + (382. + 560. i)10-s + (48.9 − 7.37i)11-s + (−44.0 − 47.4i)12-s + (2.75e3 − 2.19e3i)13-s + (1.26e3 + 48.2i)14-s + (−147. + 184. i)15-s + (124. + 1.66e3i)16-s + (2.57e3 − 8.36e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.168 + 0.429i)2-s + (−0.0474 − 0.00355i)3-s + (0.577 + 0.535i)4-s + (0.828 − 1.21i)5-s + (0.00952 − 0.0197i)6-s + (−0.329 − 0.944i)7-s + (−0.742 + 0.357i)8-s + (−0.986 − 0.148i)9-s + (0.382 + 0.560i)10-s + (0.0367 − 0.00554i)11-s + (−0.0254 − 0.0274i)12-s + (1.25 − 1.00i)13-s + (0.460 + 0.0175i)14-s + (−0.0436 + 0.0547i)15-s + (0.0304 + 0.406i)16-s + (0.524 − 1.70i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.65013 - 0.669547i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65013 - 0.669547i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (113. + 323. i)T \) |
| good | 2 | \( 1 + (1.34 - 3.43i)T + (-46.9 - 43.5i)T^{2} \) |
| 3 | \( 1 + (1.28 + 0.0960i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (-103. + 151. i)T + (-5.70e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-48.9 + 7.37i)T + (1.69e6 - 5.22e5i)T^{2} \) |
| 13 | \( 1 + (-2.75e3 + 2.19e3i)T + (1.07e6 - 4.70e6i)T^{2} \) |
| 17 | \( 1 + (-2.57e3 + 8.36e3i)T + (-1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (3.79e3 - 2.18e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-7.69e3 + 2.37e3i)T + (1.22e8 - 8.33e7i)T^{2} \) |
| 29 | \( 1 + (-3.89e3 - 1.70e4i)T + (-5.35e8 + 2.58e8i)T^{2} \) |
| 31 | \( 1 + (-4.13e4 - 2.38e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.63e3 + 2.44e3i)T + (1.91e8 - 2.55e9i)T^{2} \) |
| 41 | \( 1 + (1.96e4 + 4.07e4i)T + (-2.96e9 + 3.71e9i)T^{2} \) |
| 43 | \( 1 + (4.24e4 + 2.04e4i)T + (3.94e9 + 4.94e9i)T^{2} \) |
| 47 | \( 1 + (5.02e4 + 1.97e4i)T + (7.90e9 + 7.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.19e5 - 1.10e5i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (4.42e4 + 6.48e4i)T + (-1.54e10 + 3.92e10i)T^{2} \) |
| 61 | \( 1 + (2.00e5 + 2.15e5i)T + (-3.85e9 + 5.13e10i)T^{2} \) |
| 67 | \( 1 + (9.01e4 - 1.56e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-1.53e4 + 6.72e4i)T + (-1.15e11 - 5.55e10i)T^{2} \) |
| 73 | \( 1 + (2.84e4 - 1.11e4i)T + (1.10e11 - 1.02e11i)T^{2} \) |
| 79 | \( 1 + (-3.24e5 - 5.61e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-1.77e5 - 1.41e5i)T + (7.27e10 + 3.18e11i)T^{2} \) |
| 89 | \( 1 + (4.23e4 - 2.80e5i)T + (-4.74e11 - 1.46e11i)T^{2} \) |
| 97 | \( 1 - 5.90e5iT - 8.32e11T^{2} \) |
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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
−14.04617419818118768578665487235, −13.13181362185526014993906917322, −12.00942477436130653047698120227, −10.62770978042366186992410562967, −9.074556320550723643701848328394, −8.127684685933463769947681976486, −6.54891903586457725922167133413, −5.30212507166268837862922678978, −3.13770862581639564312298287087, −0.856032407033576547767109243788,
1.90120909557171960375362587401, 3.04911688508779257132675104649, 6.09291540886691589616719518942, 6.28656553665535772138037778539, 8.660504533002131054239317660960, 9.955417743706337491728342684367, 10.95900256662645577164482036677, 11.76320059556676364402275085139, 13.44147559271263348632818840628, 14.66243161173873654865269130295
