L(s) = 1 | + (−0.342 − 0.939i)2-s + (2.49 + 0.439i)3-s + (−0.766 + 0.642i)4-s + (−0.439 − 2.49i)6-s + (−0.565 + 0.326i)7-s + (0.866 + 0.500i)8-s + (3.20 + 1.16i)9-s + (0.5 − 0.866i)11-s + (−2.19 + 1.26i)12-s + (2.83 − 0.5i)13-s + (0.5 + 0.419i)14-s + (0.173 − 0.984i)16-s + (−0.160 − 0.439i)17-s − 3.41i·18-s + (4.07 − 1.55i)19-s + ⋯ |
L(s) = 1 | + (−0.241 − 0.664i)2-s + (1.43 + 0.253i)3-s + (−0.383 + 0.321i)4-s + (−0.179 − 1.01i)6-s + (−0.213 + 0.123i)7-s + (0.306 + 0.176i)8-s + (1.06 + 0.388i)9-s + (0.150 − 0.261i)11-s + (−0.633 + 0.365i)12-s + (0.786 − 0.138i)13-s + (0.133 + 0.112i)14-s + (0.0434 − 0.246i)16-s + (−0.0388 − 0.106i)17-s − 0.804i·18-s + (0.934 − 0.355i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.20578 - 0.598702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.20578 - 0.598702i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.07 + 1.55i)T \) |
good | 3 | \( 1 + (-2.49 - 0.439i)T + (2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (0.565 - 0.326i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.83 + 0.5i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.160 + 0.439i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.15 - 2.56i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.41 - 2.33i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.03 - 3.51i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.63iT - 37T^{2} \) |
| 41 | \( 1 + (0.854 - 4.84i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.02 + 2.40i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.19 + 6.02i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (2.48 + 2.96i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (5.68 - 2.06i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (3.96 - 3.32i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (0.300 - 0.826i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-9.35 - 7.84i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.502 + 0.0885i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.51 + 8.58i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (15.5 - 8.95i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.06 - 17.3i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (4.96 + 13.6i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−9.760080564605378455965021645224, −9.137539851654080781124305747618, −8.553182788669854591793277896172, −7.81077649861131288083027144141, −6.81214319760649907131154632231, −5.45007235083940824024273694496, −4.21830912419578165206704901488, −3.29319814708016049781274895045, −2.72359486620732901126466573544, −1.31156216178423789928539899250,
1.32727513694209879325053481170, 2.73309345598229255846344000053, 3.70336836051957133766459706157, 4.75268225915393069805631876197, 6.10338033089116145612832657691, 6.87195248281098488775930319499, 7.83372963751159063822477846167, 8.293627946889663854370597895527, 9.145990987082061377294691986902, 9.716726604100987888006059106783