L(s) = 1 | + (0.866 + 0.5i)7-s + (1.5 − 2.59i)11-s + (0.866 − 0.5i)13-s − 6i·17-s + 4·19-s + (−2.59 + 1.5i)23-s + (−1.5 + 2.59i)29-s + (−2.5 − 4.33i)31-s + 2i·37-s + (1.5 + 2.59i)41-s + (−0.866 − 0.5i)43-s + (−7.79 − 4.5i)47-s + (−3 − 5.19i)49-s − 6i·53-s + (1.5 + 2.59i)59-s + ⋯ |
L(s) = 1 | + (0.327 + 0.188i)7-s + (0.452 − 0.783i)11-s + (0.240 − 0.138i)13-s − 1.45i·17-s + 0.917·19-s + (−0.541 + 0.312i)23-s + (−0.278 + 0.482i)29-s + (−0.449 − 0.777i)31-s + 0.328i·37-s + (0.234 + 0.405i)41-s + (−0.132 − 0.0762i)43-s + (−1.13 − 0.656i)47-s + (−0.428 − 0.742i)49-s − 0.824i·53-s + (0.195 + 0.338i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.760115242\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.760115242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.866 - 0.5i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.79 + 4.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 - 3.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.79 - 4.5i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (9.52 + 5.5i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.653417634573923489201025580640, −7.999670839821001784581991155497, −7.19977647717467135593469769997, −6.42378866198112025309059830247, −5.48720345880660625204335646196, −4.95650211689555763466934952526, −3.75643694598376137151292525559, −3.06484990658246662407741500861, −1.86912232728428613208869045672, −0.61676624764186247690829387859,
1.26643466597662421878467426598, 2.14407016360217130256172821750, 3.47727373942673935744038169595, 4.18171369751995756842290522315, 5.02705321673935425612384728759, 5.98416561706554884011410879445, 6.65582595634696790783300679619, 7.57330388139866288755236047451, 8.116111436720256061783373807636, 9.032074680937160963552610888179