L(s) = 1 | + (0.409 + 1.35i)2-s − 3-s + (−1.66 + 1.10i)4-s + 0.901·5-s + (−0.409 − 1.35i)6-s + 1.56·7-s + (−2.18 − 1.79i)8-s + 9-s + (0.369 + 1.22i)10-s + 3.69i·11-s + (1.66 − 1.10i)12-s + 2.36i·13-s + (0.640 + 2.11i)14-s − 0.901·15-s + (1.53 − 3.69i)16-s + 0.473i·17-s + ⋯ |
L(s) = 1 | + (0.289 + 0.957i)2-s − 0.577·3-s + (−0.832 + 0.554i)4-s + 0.403·5-s + (−0.167 − 0.552i)6-s + 0.590·7-s + (−0.772 − 0.635i)8-s + 0.333·9-s + (0.116 + 0.385i)10-s + 1.11i·11-s + (0.480 − 0.320i)12-s + 0.656i·13-s + (0.171 + 0.565i)14-s − 0.232·15-s + (0.384 − 0.923i)16-s + 0.114i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.312344 + 1.16701i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.312344 + 1.16701i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.409 - 1.35i)T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + (4.73 - 0.790i)T \) |
good | 5 | \( 1 - 0.901T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 3.69iT - 11T^{2} \) |
| 13 | \( 1 - 2.36iT - 13T^{2} \) |
| 17 | \( 1 - 0.473iT - 17T^{2} \) |
| 19 | \( 1 - 7.66iT - 19T^{2} \) |
| 29 | \( 1 + 4.29iT - 29T^{2} \) |
| 31 | \( 1 + 1.83iT - 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 - 2.62T + 41T^{2} \) |
| 43 | \( 1 - 7.06iT - 43T^{2} \) |
| 47 | \( 1 - 4.27iT - 47T^{2} \) |
| 53 | \( 1 + 4.64T + 53T^{2} \) |
| 59 | \( 1 + 7.81T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 5.35iT - 67T^{2} \) |
| 71 | \( 1 + 6.95iT - 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 7.33iT - 83T^{2} \) |
| 89 | \( 1 + 4.91iT - 89T^{2} \) |
| 97 | \( 1 + 15.0iT - 97T^{2} \) |
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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
−11.31590962716386347606364017464, −9.885398075990138139429729067057, −9.606541242689953978721727411745, −8.088922502720785960299671165676, −7.62368710451987551627222254418, −6.37279603678028952775984990260, −5.82492134681746908998923950419, −4.68667834781785390276280844414, −3.96530091591541341848288870925, −1.87317105255760159619306593625,
0.70068324607978537807055156499, 2.24986472558826696881243627627, 3.53109873252411897107900147510, 4.80628383036519509327570973373, 5.52456363925635011424065344136, 6.44318361150527337469965973884, 7.934825270998779249446868875264, 8.873068346943549404418302491927, 9.764196155353855455442762074801, 10.73997224414340892932033281592