L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.382 − 0.923i)3-s + (−0.866 − 0.499i)4-s + (−0.923 − 0.382i)5-s + (0.991 − 0.130i)6-s + (0.965 + 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.608 − 0.793i)10-s + (−0.130 + 0.991i)12-s + (0.739 − 0.198i)13-s + (−0.499 + 0.866i)14-s + i·15-s + (0.500 + 0.866i)16-s + (−0.500 − 0.866i)18-s − 0.261i·19-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.382 − 0.923i)3-s + (−0.866 − 0.499i)4-s + (−0.923 − 0.382i)5-s + (0.991 − 0.130i)6-s + (0.965 + 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.608 − 0.793i)10-s + (−0.130 + 0.991i)12-s + (0.739 − 0.198i)13-s + (−0.499 + 0.866i)14-s + i·15-s + (0.500 + 0.866i)16-s + (−0.500 − 0.866i)18-s − 0.261i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7464693269\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7464693269\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 + (0.382 + 0.923i)T \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 + (-0.965 - 0.258i)T \) |
good | 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.739 + 0.198i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 0.261iT - T^{2} \) |
| 23 | \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + 1.93iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
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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
−8.618159140106370480989300984313, −8.069939237399828439058833406548, −7.70109325307319113264292624698, −6.78631457216355814785258336465, −6.11177876375817732162205212423, −5.15364067456802435951175705017, −4.68430718770571630818271622403, −3.52156619576612235286721814370, −1.86820169698814064000117125532, −0.72321670365527993664146566050,
1.13651310756066894116329087623, 2.62222006087666582939404022932, 3.70586448304964562468035629544, 4.09913901427167676237283628604, 4.90722960712440956022159039944, 5.78647089524455355761587266206, 7.05174860975355057576084051741, 7.936788071424043272453523910206, 8.538526960822909879729880589425, 9.197218724288004002622526209993