L(s) = 1 | + (−1 + 1.73i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s + (−2 − 3.46i)5-s − 3.46i·6-s + (−1 + 1.73i)7-s + (1.5 − 2.59i)9-s + 7.99·10-s + (−1 + 1.73i)11-s + (2.99 + 1.73i)12-s + (−0.5 − 0.866i)13-s + (−1.99 − 3.46i)14-s + (6 + 3.46i)15-s + (1.99 − 3.46i)16-s − 5·17-s + (3 + 5.19i)18-s + ⋯ |
L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.866 + 0.499i)3-s + (−0.499 − 0.866i)4-s + (−0.894 − 1.54i)5-s − 1.41i·6-s + (−0.377 + 0.654i)7-s + (0.5 − 0.866i)9-s + 2.52·10-s + (−0.301 + 0.522i)11-s + (0.866 + 0.500i)12-s + (−0.138 − 0.240i)13-s + (−0.534 − 0.925i)14-s + (1.54 + 0.894i)15-s + (0.499 − 0.866i)16-s − 1.21·17-s + (0.707 + 1.22i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (1 - 1.73i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 2.59i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (-4 + 6.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.88404769380007274792023612784, −12.34412890230565534100741464708, −11.19543785121556848574998822614, −9.567657908037077911008199632307, −8.891939893687426618524028996333, −7.905208079325265537173997537277, −6.54890166747572833456200045986, −5.38590008515613444261790887688, −4.36470287146865125139193462229, 0,
2.50964273036828341017701981849, 3.99955899920403288461230269357, 6.39603500228795107885277515346, 7.16409432509687099870075453860, 8.563269284622916452625217341821, 10.39039528355302913521713267168, 10.71820635543410582069665704574, 11.38179457796890927224920857107, 12.36886985559743262914532758057