| L(s) = 1 | + (−0.415 − 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (2.06 + 1.32i)5-s + (0.654 + 0.755i)6-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (0.349 − 2.43i)10-s + (0.555 − 1.21i)11-s + (0.415 − 0.909i)12-s + (0.0102 − 0.0711i)13-s + (−0.841 + 0.540i)14-s + (−2.35 − 0.692i)15-s + (−0.142 − 0.989i)16-s + (−1.88 − 2.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.924 + 0.593i)5-s + (0.267 + 0.308i)6-s + (−0.0537 − 0.374i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.110 − 0.768i)10-s + (0.167 − 0.366i)11-s + (0.119 − 0.262i)12-s + (0.00283 − 0.0197i)13-s + (−0.224 + 0.144i)14-s + (−0.608 − 0.178i)15-s + (−0.0355 − 0.247i)16-s + (−0.458 − 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23654 - 0.419234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23654 - 0.419234i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-4.51 - 1.61i)T \) |
| good | 5 | \( 1 + (-2.06 - 1.32i)T + (2.07 + 4.54i)T^{2} \) |
| 11 | \( 1 + (-0.555 + 1.21i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.0102 + 0.0711i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (1.88 + 2.18i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.17 + 1.35i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-1.94 - 2.24i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-8.06 - 2.36i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (7.27 - 4.67i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (4.03 + 2.59i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-9.61 + 2.82i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + (-0.145 - 1.01i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.84 + 12.8i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-13.3 - 3.93i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-0.848 - 1.85i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (4.29 + 9.39i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (6.76 - 7.80i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.747 - 5.20i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-7.67 + 4.93i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-7.76 + 2.28i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (9.89 + 6.35i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.18171819609252763543556004127, −9.308309498176031368045505151907, −8.558332931350611894370630820503, −7.20370269829688965503573892760, −6.61233737176936012141102923413, −5.54955138031536867878174462801, −4.63736958505507329790972334170, −3.39031619272085242733069226055, −2.39167950163291495528027078072, −0.949030316573601719518930288290,
1.08277127330200097001452395104, 2.34339702857136179724066407319, 4.21021049704496672966364675195, 5.16515370618706691464277580211, 5.85532998853147942618233507699, 6.57175754325400190344888481164, 7.48614527013708325947042477861, 8.578604797853102330380541665152, 9.167328821627925222310011779961, 9.999309345522464100611846815985
