| L(s) = 1 | + (234. + 406. i)2-s + (−4.04e3 − 2.33e3i)3-s + (−7.73e4 + 1.33e5i)4-s + (5.90e4 − 3.40e4i)5-s − 2.19e6i·6-s + (−5.75e6 + 3.38e5i)7-s − 4.18e7·8-s + (−1.06e7 − 1.83e7i)9-s + (2.76e7 + 1.59e7i)10-s + (1.14e8 − 1.98e8i)11-s + (6.25e8 − 3.61e8i)12-s + 1.08e9i·13-s + (−1.48e9 − 2.25e9i)14-s − 3.18e8·15-s + (−4.74e9 − 8.22e9i)16-s + (−6.78e8 − 3.91e8i)17-s + ⋯ |
| L(s) = 1 | + (0.916 + 1.58i)2-s + (−0.616 − 0.355i)3-s + (−1.18 + 2.04i)4-s + (0.151 − 0.0872i)5-s − 1.30i·6-s + (−0.998 + 0.0586i)7-s − 2.49·8-s + (−0.246 − 0.427i)9-s + (0.276 + 0.159i)10-s + (0.534 − 0.925i)11-s + (1.45 − 0.840i)12-s + 1.33i·13-s + (−1.00 − 1.53i)14-s − 0.124·15-s + (−1.10 − 1.91i)16-s + (−0.0972 − 0.0561i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(0.359720 - 0.507810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.359720 - 0.507810i\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (5.75e6 - 3.38e5i)T \) |
| good | 2 | \( 1 + (-234. - 406. i)T + (-3.27e4 + 5.67e4i)T^{2} \) |
| 3 | \( 1 + (4.04e3 + 2.33e3i)T + (2.15e7 + 3.72e7i)T^{2} \) |
| 5 | \( 1 + (-5.90e4 + 3.40e4i)T + (7.62e10 - 1.32e11i)T^{2} \) |
| 11 | \( 1 + (-1.14e8 + 1.98e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 13 | \( 1 - 1.08e9iT - 6.65e17T^{2} \) |
| 17 | \( 1 + (6.78e8 + 3.91e8i)T + (2.43e19 + 4.21e19i)T^{2} \) |
| 19 | \( 1 + (2.45e10 - 1.41e10i)T + (1.44e20 - 2.49e20i)T^{2} \) |
| 23 | \( 1 + (4.24e10 + 7.34e10i)T + (-3.06e21 + 5.31e21i)T^{2} \) |
| 29 | \( 1 + 5.71e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + (-7.61e11 - 4.39e11i)T + (3.63e23 + 6.29e23i)T^{2} \) |
| 37 | \( 1 + (-9.66e11 - 1.67e12i)T + (-6.16e24 + 1.06e25i)T^{2} \) |
| 41 | \( 1 - 4.03e12iT - 6.37e25T^{2} \) |
| 43 | \( 1 + 3.48e12T + 1.36e26T^{2} \) |
| 47 | \( 1 + (-2.40e13 + 1.38e13i)T + (2.83e26 - 4.91e26i)T^{2} \) |
| 53 | \( 1 + (-2.76e13 + 4.78e13i)T + (-1.93e27 - 3.35e27i)T^{2} \) |
| 59 | \( 1 + (-1.32e14 - 7.65e13i)T + (1.07e28 + 1.86e28i)T^{2} \) |
| 61 | \( 1 + (1.82e14 - 1.05e14i)T + (1.83e28 - 3.18e28i)T^{2} \) |
| 67 | \( 1 + (-1.30e14 + 2.26e14i)T + (-8.24e28 - 1.42e29i)T^{2} \) |
| 71 | \( 1 + 6.97e14T + 4.16e29T^{2} \) |
| 73 | \( 1 + (-3.30e14 - 1.90e14i)T + (3.25e29 + 5.63e29i)T^{2} \) |
| 79 | \( 1 + (-1.62e14 - 2.82e14i)T + (-1.15e30 + 1.99e30i)T^{2} \) |
| 83 | \( 1 + 2.91e15iT - 5.07e30T^{2} \) |
| 89 | \( 1 + (-1.17e15 + 6.79e14i)T + (7.74e30 - 1.34e31i)T^{2} \) |
| 97 | \( 1 - 6.22e15iT - 6.14e31T^{2} \) |
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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
−18.81119950826311702840075784586, −17.00712657723350252317044381536, −16.42990101426561397292324614764, −14.73025169610344520314141952980, −13.37524041436729108115471606805, −12.04319829141807917036322729686, −8.833120301049027133540245959531, −6.66289438851501820095609415496, −5.97961695936920321343968736345, −3.87376435884968080832092176706,
0.21369555777729619615437382577, 2.41271351870158325258435268423, 4.17250761196867934862058014294, 5.80580494136790279118111901592, 9.783267747036050598730648167788, 10.82609676499944527975181632536, 12.32609725424609830641442875450, 13.46001624805263839493895694199, 15.28145547719705821137415766531, 17.46783021204901307449396331513
