L(s) = 1 | + (−1.22 + 0.707i)2-s + (−2.72 + 1.57i)3-s + (0.999 − 1.73i)4-s + (0.5 − 0.866i)5-s + (2.22 − 3.85i)6-s + (2.5 + 0.866i)7-s + 2.82i·8-s + (3.44 − 5.97i)9-s + 1.41i·10-s + (2.44 + 4.24i)11-s + 6.29i·12-s − 0.449·13-s + (−3.67 + 0.707i)14-s + 3.14i·15-s + (−2.00 − 3.46i)16-s + (−1.77 + 1.02i)17-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)2-s + (−1.57 + 0.908i)3-s + (0.499 − 0.866i)4-s + (0.223 − 0.387i)5-s + (0.908 − 1.57i)6-s + (0.944 + 0.327i)7-s + 0.999i·8-s + (1.14 − 1.99i)9-s + 0.447i·10-s + (0.738 + 1.27i)11-s + 1.81i·12-s − 0.124·13-s + (−0.981 + 0.188i)14-s + 0.812i·15-s + (−0.500 − 0.866i)16-s + (−0.430 + 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.251767 + 0.469850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.251767 + 0.469850i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 3 | \( 1 + (2.72 - 1.57i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 + (1.77 - 1.02i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.67 + 2.12i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.949 - 0.548i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 10.0iT - 29T^{2} \) |
| 31 | \( 1 + (-3.22 - 5.58i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 - 1.73i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.36iT - 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.22 + 2.43i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (9.12 - 5.26i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.17 + 12.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.17 - 2.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 + (2.32 - 1.34i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.674 + 0.389i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 + (0.398 + 0.230i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.02iT - 97T^{2} \) |
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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
−11.82200951542367474190375196333, −11.04953111226226761175726710088, −10.32421903596451213898741706233, −9.430621613898929041223202094539, −8.619261602319025562529121165025, −7.05438092233608930529434958414, −6.26970709892000490355201848940, −5.03674933363583516986998518936, −4.61924441858115855013987352411, −1.51120056911548877095012953310,
0.70659462693298453991776413315, 2.08550631939329448872035821156, 4.24514754222682582703660249740, 5.87777112152146549796695598733, 6.59914667291269252852283189549, 7.62497987485950307595613875208, 8.493897877799311899413935143154, 10.01643709815444322276537021957, 10.91587047832998234563342569651, 11.44962529304273471909483521657