| L(s) = 1 | + (−0.220 − 0.909i)2-s + (−0.294 + 2.05i)3-s + (0.999 − 0.515i)4-s + (0.408 + 0.894i)5-s + (1.93 − 0.184i)6-s + (0.776 − 0.740i)7-s + (−1.91 − 2.20i)8-s + (−1.24 − 0.364i)9-s + (0.723 − 0.569i)10-s + (−3.75 − 0.358i)11-s + (0.762 + 2.20i)12-s + (−2.49 − 0.480i)13-s + (−0.844 − 0.543i)14-s + (−1.95 + 0.574i)15-s + (−0.282 + 0.396i)16-s + (1.73 + 0.896i)17-s + ⋯ |
| L(s) = 1 | + (−0.155 − 0.643i)2-s + (−0.170 + 1.18i)3-s + (0.499 − 0.257i)4-s + (0.182 + 0.400i)5-s + (0.788 − 0.0752i)6-s + (0.293 − 0.279i)7-s + (−0.676 − 0.781i)8-s + (−0.413 − 0.121i)9-s + (0.228 − 0.179i)10-s + (−1.13 − 0.107i)11-s + (0.219 + 0.635i)12-s + (−0.691 − 0.133i)13-s + (−0.225 − 0.145i)14-s + (−0.505 + 0.148i)15-s + (−0.0706 + 0.0991i)16-s + (0.421 + 0.217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.894547 - 0.0116961i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.894547 - 0.0116961i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (6.97 + 4.28i)T \) |
| good | 2 | \( 1 + (0.220 + 0.909i)T + (-1.77 + 0.916i)T^{2} \) |
| 3 | \( 1 + (0.294 - 2.05i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (-0.408 - 0.894i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.776 + 0.740i)T + (0.333 - 6.99i)T^{2} \) |
| 11 | \( 1 + (3.75 + 0.358i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (2.49 + 0.480i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-1.73 - 0.896i)T + (9.86 + 13.8i)T^{2} \) |
| 19 | \( 1 + (1.71 + 1.63i)T + (0.904 + 18.9i)T^{2} \) |
| 23 | \( 1 + (7.22 - 2.89i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (-3.52 + 6.09i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-9.09 + 1.75i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (-2.95 - 5.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0149 - 0.314i)T + (-40.8 + 3.89i)T^{2} \) |
| 43 | \( 1 + (7.48 - 4.80i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-8.60 - 6.76i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (-2.47 - 1.59i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (0.832 + 0.960i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (6.38 - 0.609i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (-3.34 + 1.72i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (-10.4 + 0.997i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (-3.12 - 9.01i)T + (-62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (-3.18 + 4.46i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (0.500 + 3.48i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (8.87 + 15.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−15.18720752798136205815919142291, −13.82990849186408459885821449197, −12.27515526715699070711609091665, −11.13085794790633368849944917137, −10.21097255622036470506295377145, −9.855065881576762566375279617369, −7.87004110718113894171260286230, −6.12180529809893910386556138549, −4.54999265707008805688028080832, −2.74729279405573491802223421686,
2.29511030769388896914868398477, 5.32070824420367829438916801306, 6.60640225996402955496775106271, 7.65081285317183180009460440089, 8.477793093726597036040466482030, 10.34740753758152983319354187973, 12.01113684000005291463249681605, 12.42144004926619278230629912644, 13.69562765648858233060699671520, 14.92259537843333292968728817899
