| L(s) = 1 | + (7.75 − 7.19i)2-s + (−22.4 + 3.38i)3-s + (5.96 − 79.6i)4-s + (−24.3 + 62.0i)5-s + (−149. + 187. i)6-s + (−121. + 46.3i)7-s + (−315. − 395. i)8-s + (259. − 80.0i)9-s + (257. + 656. i)10-s + (90.5 + 27.9i)11-s + (135. + 1.80e3i)12-s + (−185. − 810. i)13-s + (−605. + 1.23e3i)14-s + (336. − 1.47e3i)15-s + (−2.76e3 − 416. i)16-s + (−1.63e3 + 1.11e3i)17-s + ⋯ |
| L(s) = 1 | + (1.37 − 1.27i)2-s + (−1.43 + 0.216i)3-s + (0.186 − 2.48i)4-s + (−0.435 + 1.11i)5-s + (−1.69 + 2.12i)6-s + (−0.933 + 0.357i)7-s + (−1.74 − 2.18i)8-s + (1.06 − 0.329i)9-s + (0.814 + 2.07i)10-s + (0.225 + 0.0696i)11-s + (0.271 + 3.62i)12-s + (−0.303 − 1.33i)13-s + (−0.825 + 1.67i)14-s + (0.386 − 1.69i)15-s + (−2.70 − 0.407i)16-s + (−1.37 + 0.934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.151163 + 0.269688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151163 + 0.269688i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (121. - 46.3i)T \) |
| good | 2 | \( 1 + (-7.75 + 7.19i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (22.4 - 3.38i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (24.3 - 62.0i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (-90.5 - 27.9i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (185. + 810. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (1.63e3 - 1.11e3i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (431. + 748. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (477. + 325. i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (5.16e3 + 2.48e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-3.98e3 + 6.89e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-902. - 1.20e4i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (1.01e3 + 1.27e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-4.84e3 + 6.08e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (4.47e3 - 4.14e3i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (1.97e3 - 2.63e4i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (5.43e3 + 1.38e4i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (39.3 + 525. i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (1.29e4 - 2.23e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (8.42e3 - 4.05e3i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-2.84e4 - 2.63e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (3.49e4 + 6.05e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.65e3 - 1.59e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (7.53e4 - 2.32e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 + 1.23e5T + 8.58e9T^{2} \) |
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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
−13.34097454960806310734932956081, −12.50843953278618287142631881680, −11.46296633004298047810095255056, −10.80119861785158954939744138405, −9.965521866721115554015869357452, −6.56063289668334912224357612453, −5.76570568719477456245039133235, −4.21358750653796915901342947237, −2.76368321239211256512686696591, −0.11530698111645435928212119395,
4.14928337450504161939484350523, 5.07056627589267427764927305874, 6.37792865937350600172522935223, 7.13373143154672294657378418937, 8.973669001208690833312917295469, 11.40242215412566983892737418404, 12.30846973763023613691962304294, 12.96358836673683427575153380795, 14.09796539746505586815933209015, 15.76631905958869386580081456296
