L(s) = 1 | − 2.35·2-s − 3-s + 3.55·4-s − 3.59·5-s + 2.35·6-s − 4.00·7-s − 3.67·8-s + 9-s + 8.48·10-s − 3.87·11-s − 3.55·12-s + 9.45·14-s + 3.59·15-s + 1.54·16-s − 17-s − 2.35·18-s + 4.91·19-s − 12.8·20-s + 4.00·21-s + 9.14·22-s + 4.25·23-s + 3.67·24-s + 7.94·25-s − 27-s − 14.2·28-s + 7.32·29-s − 8.48·30-s + ⋯ |
L(s) = 1 | − 1.66·2-s − 0.577·3-s + 1.77·4-s − 1.60·5-s + 0.962·6-s − 1.51·7-s − 1.29·8-s + 0.333·9-s + 2.68·10-s − 1.16·11-s − 1.02·12-s + 2.52·14-s + 0.928·15-s + 0.385·16-s − 0.242·17-s − 0.555·18-s + 1.12·19-s − 2.86·20-s + 0.874·21-s + 1.94·22-s + 0.887·23-s + 0.749·24-s + 1.58·25-s − 0.192·27-s − 2.69·28-s + 1.35·29-s − 1.54·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2055952186\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2055952186\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 5 | \( 1 + 3.59T + 5T^{2} \) |
| 7 | \( 1 + 4.00T + 7T^{2} \) |
| 11 | \( 1 + 3.87T + 11T^{2} \) |
| 19 | \( 1 - 4.91T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 - 7.32T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 - 0.334T + 43T^{2} \) |
| 47 | \( 1 + 6.18T + 47T^{2} \) |
| 53 | \( 1 + 7.45T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 + 5.20T + 67T^{2} \) |
| 71 | \( 1 + 6.71T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78771316275652055337420276242, −7.34737568496037326623548035061, −6.67389994705505171918321749572, −6.12738783482641587396405621781, −4.97332590823461600691285157441, −4.24428532559544493189793012510, −2.99754021998403748352789746614, −2.84905830301024502456615079906, −1.06369720470055236966239940568, −0.36365083805269719141065113422,
0.36365083805269719141065113422, 1.06369720470055236966239940568, 2.84905830301024502456615079906, 2.99754021998403748352789746614, 4.24428532559544493189793012510, 4.97332590823461600691285157441, 6.12738783482641587396405621781, 6.67389994705505171918321749572, 7.34737568496037326623548035061, 7.78771316275652055337420276242