| L(s) = 1 | + (−0.451 + 1.34i)2-s + (−2.91 − 0.721i)3-s + (−1.59 − 1.21i)4-s + (8.94 + 3.56i)5-s + (2.28 − 3.57i)6-s + (−3.17 + 1.46i)7-s + (2.34 − 1.58i)8-s + (7.95 + 4.20i)9-s + (−8.81 + 10.3i)10-s + (−2.20 − 0.611i)11-s + (3.76 + 4.67i)12-s + (7.92 − 14.9i)13-s + (−0.535 − 4.92i)14-s + (−23.4 − 16.8i)15-s + (1.07 + 3.85i)16-s + (0.0763 − 0.164i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.670i)2-s + (−0.970 − 0.240i)3-s + (−0.398 − 0.302i)4-s + (1.78 + 0.712i)5-s + (0.380 − 0.596i)6-s + (−0.453 + 0.209i)7-s + (0.292 − 0.198i)8-s + (0.884 + 0.466i)9-s + (−0.881 + 1.03i)10-s + (−0.200 − 0.0555i)11-s + (0.313 + 0.389i)12-s + (0.609 − 1.14i)13-s + (−0.0382 − 0.351i)14-s + (−1.56 − 1.12i)15-s + (0.0668 + 0.240i)16-s + (0.00449 − 0.00970i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13015 + 0.818322i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13015 + 0.818322i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.451 - 1.34i)T \) |
| 3 | \( 1 + (2.91 + 0.721i)T \) |
| 59 | \( 1 + (55.5 - 19.9i)T \) |
| good | 5 | \( 1 + (-8.94 - 3.56i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (3.17 - 1.46i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (2.20 + 0.611i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-7.92 + 14.9i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (-0.0763 + 0.164i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-7.06 + 1.55i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-16.6 - 2.72i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-11.9 - 35.3i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (2.36 + 0.520i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (27.1 - 39.9i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (13.5 - 2.22i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-17.3 - 62.5i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-75.9 + 30.2i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-55.1 + 46.8i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-80.7 - 27.2i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (6.31 + 9.31i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (34.2 - 13.6i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-46.1 + 5.01i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-0.162 + 0.0976i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (48.9 - 2.65i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (28.4 + 84.5i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (138. + 15.0i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−11.09828184910900232134185391209, −10.36394075458326993860612058932, −9.815143855839180289499715013638, −8.715966040009397925542682299095, −7.24247323250148773852062148483, −6.50088456028348168915983595950, −5.73217845694621430474206852583, −5.16006359444698521025727652043, −2.95792723605215312010874257755, −1.27249369229756446034981541083,
0.943298261508357333033831138321, 2.16628938521314215761600561745, 4.06357978160014137981637287707, 5.19571946482319653506748265269, 6.01552575541749391234486201452, 6.97517802883135591007803958173, 8.841954917743385577582751373392, 9.423579382839731639803161979827, 10.18190986035621728859703889053, 10.86784062062298648734574606677
