L(s) = 1 | + (0.866 − 1.11i)2-s + 1.73·3-s + (−0.500 − 1.93i)4-s + (1.49 − 1.93i)6-s + (−2.59 − 1.11i)8-s + 2.99·9-s + (−0.866 − 3.35i)12-s + (−3.5 + 1.93i)16-s − 4.47i·17-s + (2.59 − 3.35i)18-s + 7.74i·19-s − 3.46·23-s + (−4.50 − 1.93i)24-s + 5.19·27-s + 7.74i·31-s + (−0.866 + 5.59i)32-s + ⋯ |
L(s) = 1 | + (0.612 − 0.790i)2-s + 1.00·3-s + (−0.250 − 0.968i)4-s + (0.612 − 0.790i)6-s + (−0.918 − 0.395i)8-s + 0.999·9-s + (−0.250 − 0.968i)12-s + (−0.875 + 0.484i)16-s − 1.08i·17-s + (0.612 − 0.790i)18-s + 1.77i·19-s − 0.722·23-s + (−0.918 − 0.395i)24-s + 1.00·27-s + 1.39i·31-s + (−0.153 + 0.988i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78369 - 1.38164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78369 - 1.38164i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 1.11i)T \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 7.74iT - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 7.74iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 7.74iT - 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.78175675085698842844272736484, −10.46947671026963346654234935341, −9.843221995395319042573180136518, −8.901599818302637925059394638066, −7.83743197920996102622876190385, −6.56238621750688459368607035467, −5.21252859413873842000939297548, −4.00566856173883031188541835477, −3.03187788127953670040836800673, −1.69383047641562142343578541543,
2.45657223494028021055502165666, 3.74233123488470895105617752460, 4.70492407682606185113424540184, 6.12363996841608109918257343483, 7.14311312029625923470584536406, 8.042117752955831477681839279600, 8.834857352242505299862297906693, 9.753943823661843734144148892787, 11.11465149098388891745483785302, 12.29659070470661254217262546053