| L(s) = 1 | + (1.66 − 1.30i)2-s + (−0.814 + 0.580i)3-s + (0.586 − 2.41i)4-s + (2.07 + 0.399i)5-s + (−0.596 + 2.03i)6-s + (−0.0453 + 2.64i)7-s + (−0.428 − 0.937i)8-s + (0.327 − 0.945i)9-s + (3.97 − 2.04i)10-s + (2.42 − 3.07i)11-s + (0.924 + 2.30i)12-s + (−0.763 + 1.18i)13-s + (3.38 + 4.46i)14-s + (−1.91 + 0.876i)15-s + (2.47 + 1.27i)16-s + (−0.200 + 0.191i)17-s + ⋯ |
| L(s) = 1 | + (1.17 − 0.925i)2-s + (−0.470 + 0.334i)3-s + (0.293 − 1.20i)4-s + (0.926 + 0.178i)5-s + (−0.243 + 0.829i)6-s + (−0.0171 + 0.999i)7-s + (−0.151 − 0.331i)8-s + (0.109 − 0.315i)9-s + (1.25 − 0.647i)10-s + (0.730 − 0.928i)11-s + (0.266 + 0.666i)12-s + (−0.211 + 0.329i)13-s + (0.905 + 1.19i)14-s + (−0.495 + 0.226i)15-s + (0.619 + 0.319i)16-s + (−0.0486 + 0.0464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.53392 - 0.767205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.53392 - 0.767205i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 - 0.580i)T \) |
| 7 | \( 1 + (0.0453 - 2.64i)T \) |
| 23 | \( 1 + (-2.38 + 4.15i)T \) |
| good | 2 | \( 1 + (-1.66 + 1.30i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-2.07 - 0.399i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (-2.42 + 3.07i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (0.763 - 1.18i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.200 - 0.191i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.417 - 0.397i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (5.20 + 1.52i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.233 - 2.44i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (3.04 + 1.05i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (4.70 + 4.07i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.42 + 1.56i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-1.68 + 0.971i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.34 - 0.159i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-3.02 - 5.86i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (4.91 - 6.90i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-3.71 + 9.28i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.43 - 10.0i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.40 - 0.583i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (9.79 + 0.466i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-8.30 - 9.58i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (12.8 - 1.22i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-11.3 + 13.0i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.13774121013971768469519987364, −10.32564834414763108483045329335, −9.380265070493060105814209561069, −8.522874346128035604176487934040, −6.65368444529467347270927768750, −5.79688759921568284299881173163, −5.24374540756339425070761175205, −4.02289846066562389327699381858, −2.90806739193934858360336748481, −1.78998637764941082432997318984,
1.57061811225511440042490852173, 3.55283689153724489209188421111, 4.67431037885905159808900369317, 5.41506485377483212495226284533, 6.39293379365013946325040233356, 7.06443711247259911687894214965, 7.79125620026098491625734274136, 9.451672987510876205314221787059, 10.09295388116466893727022581986, 11.27574302976485142174468674166
