L(s) = 1 | + (1 − 1.73i)4-s + (0.5 − 0.866i)5-s + (2 + 3.46i)7-s + (−0.5 + 0.866i)13-s + (−1.99 − 3.46i)16-s + 3·17-s + 2·19-s + (−0.999 − 1.73i)20-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s + 7.99·28-s + (3 + 5.19i)29-s + (2 − 3.46i)31-s + 3.99·35-s + 8·37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (0.223 − 0.387i)5-s + (0.755 + 1.30i)7-s + (−0.138 + 0.240i)13-s + (−0.499 − 0.866i)16-s + 0.727·17-s + 0.458·19-s + (−0.223 − 0.387i)20-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s + 1.51·28-s + (0.557 + 0.964i)29-s + (0.359 − 0.622i)31-s + 0.676·35-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.330992472\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.330992472\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + (-3.5 - 6.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.403155876338105676392269373312, −8.487571034533226330828972846579, −7.81681028651080636487855266106, −6.66130518890771556708260954790, −5.96760088895347198077761988271, −5.18890188477086273964544189886, −4.69559368287568379232720777869, −2.99940436077689140365389986007, −2.10070271239559899522692348387, −1.13384088460813963608273686864,
1.13313528379612462835376192986, 2.43863084164657004830108658955, 3.45314468259289651694213868611, 4.18429648534170975266675228094, 5.22231586328429538434943673418, 6.32873195327169383044113135816, 7.22720525505840888932543939523, 7.61943988383422917704698316668, 8.296089072329112870747647836442, 9.357392785698868320497874898150