| L(s) = 1 | + (2.04 + 1.03i)2-s + (2.07 − 1.96i)3-s + (0.743 + 1.01i)4-s + (−0.325 + 0.00347i)5-s + (6.29 − 1.86i)6-s + (4.62 − 5.25i)7-s + (−1.00 − 6.20i)8-s + (−0.0436 + 0.816i)9-s + (−0.669 − 0.331i)10-s + (3.54 − 3.15i)11-s + (3.54 + 0.652i)12-s + (−5.61 − 3.40i)13-s + (14.9 − 5.92i)14-s + (−0.668 + 0.647i)15-s + (5.93 − 18.5i)16-s + (6.57 − 0.776i)17-s + ⋯ |
| L(s) = 1 | + (1.02 + 0.519i)2-s + (0.691 − 0.655i)3-s + (0.185 + 0.254i)4-s + (−0.0650 + 0.000695i)5-s + (1.04 − 0.311i)6-s + (0.661 − 0.750i)7-s + (−0.125 − 0.775i)8-s + (−0.00485 + 0.0907i)9-s + (−0.0669 − 0.0331i)10-s + (0.322 − 0.286i)11-s + (0.295 + 0.0543i)12-s + (−0.431 − 0.261i)13-s + (1.06 − 0.423i)14-s + (−0.0445 + 0.0431i)15-s + (0.370 − 1.15i)16-s + (0.386 − 0.0456i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.31477 - 1.05377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.31477 - 1.05377i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-4.62 + 5.25i)T \) |
| good | 2 | \( 1 + (-2.04 - 1.03i)T + (2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (-2.07 + 1.96i)T + (0.480 - 8.98i)T^{2} \) |
| 5 | \( 1 + (0.325 - 0.00347i)T + (24.9 - 0.534i)T^{2} \) |
| 11 | \( 1 + (-3.54 + 3.15i)T + (14.1 - 120. i)T^{2} \) |
| 13 | \( 1 + (5.61 + 3.40i)T + (78.1 + 149. i)T^{2} \) |
| 17 | \( 1 + (-6.57 + 0.776i)T + (281. - 67.3i)T^{2} \) |
| 19 | \( 1 + (-3.20 + 1.85i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.46 - 18.7i)T + (-417. + 325. i)T^{2} \) |
| 29 | \( 1 + (-17.5 + 3.42i)T + (779. - 315. i)T^{2} \) |
| 31 | \( 1 + (-3.81 + 1.49i)T + (704. - 653. i)T^{2} \) |
| 37 | \( 1 + (-29.9 - 12.8i)T + (941. + 993. i)T^{2} \) |
| 41 | \( 1 + (-7.43 + 5.18i)T + (580. - 1.57e3i)T^{2} \) |
| 43 | \( 1 + (6.70 - 41.4i)T + (-1.75e3 - 582. i)T^{2} \) |
| 47 | \( 1 + (10.1 + 2.55i)T + (1.94e3 + 1.04e3i)T^{2} \) |
| 53 | \( 1 + (23.8 - 86.9i)T + (-2.41e3 - 1.43e3i)T^{2} \) |
| 59 | \( 1 + (41.9 - 19.6i)T + (2.22e3 - 2.67e3i)T^{2} \) |
| 61 | \( 1 + (0.517 + 12.0i)T + (-3.70e3 + 317. i)T^{2} \) |
| 67 | \( 1 + (63.7 - 9.61i)T + (4.28e3 - 1.32e3i)T^{2} \) |
| 71 | \( 1 + (56.3 + 10.9i)T + (4.67e3 + 1.89e3i)T^{2} \) |
| 73 | \( 1 + (-8.43 + 2.11i)T + (4.69e3 - 2.51e3i)T^{2} \) |
| 79 | \( 1 + (-10.0 - 134. i)T + (-6.17e3 + 930. i)T^{2} \) |
| 83 | \( 1 + (-70.2 + 6.77i)T + (6.76e3 - 1.31e3i)T^{2} \) |
| 89 | \( 1 + (-108. + 45.4i)T + (5.57e3 - 5.63e3i)T^{2} \) |
| 97 | \( 1 + (-97.8 - 78.0i)T + (2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.46299241417890026547521771285, −10.30060594529673394572375873877, −9.255070039134982894093308122938, −7.88334531032242739241467340889, −7.48412391028672048424719486188, −6.34514087904389369007562070224, −5.19530247372481796867610197023, −4.22467353790733236172868008244, −3.00494912032302716638876040530, −1.25834069711905379270084859835,
2.12481453767692849300315900356, 3.19506944929028249016414742466, 4.24491243816655827096871347564, 5.01362187631650249325285680756, 6.21451482232900183116336297238, 7.84685347389046896040579081992, 8.706082765532563702299365983697, 9.504109667199298865457583606835, 10.60475143724540750315415377656, 11.82679776564672546784887500651
