L(s) = 1 | + (0.327 − 0.945i)3-s + (−0.723 + 0.690i)4-s + (−0.142 − 0.989i)7-s + (−0.786 − 0.618i)9-s + (0.415 + 0.909i)12-s + (−0.283 − 0.0270i)13-s + (0.0475 − 0.998i)16-s + (−1.16 − 1.48i)19-s + (−0.981 − 0.189i)21-s + (0.580 − 0.814i)25-s + (−0.841 + 0.540i)27-s + (0.786 + 0.618i)28-s + (0.975 + 1.37i)31-s + (0.995 − 0.0950i)36-s + (−0.580 − 1.00i)37-s + ⋯ |
L(s) = 1 | + (0.327 − 0.945i)3-s + (−0.723 + 0.690i)4-s + (−0.142 − 0.989i)7-s + (−0.786 − 0.618i)9-s + (0.415 + 0.909i)12-s + (−0.283 − 0.0270i)13-s + (0.0475 − 0.998i)16-s + (−1.16 − 1.48i)19-s + (−0.981 − 0.189i)21-s + (0.580 − 0.814i)25-s + (−0.841 + 0.540i)27-s + (0.786 + 0.618i)28-s + (0.975 + 1.37i)31-s + (0.995 − 0.0950i)36-s + (−0.580 − 1.00i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7821629801\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7821629801\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.327 + 0.945i)T \) |
| 7 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (0.415 + 0.909i)T \) |
good | 2 | \( 1 + (0.723 - 0.690i)T^{2} \) |
| 5 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 11 | \( 1 + (0.580 - 0.814i)T^{2} \) |
| 13 | \( 1 + (0.283 + 0.0270i)T + (0.981 + 0.189i)T^{2} \) |
| 17 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (1.16 + 1.48i)T + (-0.235 + 0.971i)T^{2} \) |
| 23 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.975 - 1.37i)T + (-0.327 + 0.945i)T^{2} \) |
| 37 | \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 43 | \( 1 + (0.512 + 1.74i)T + (-0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 53 | \( 1 + (-0.888 + 0.458i)T^{2} \) |
| 59 | \( 1 + (0.981 - 0.189i)T^{2} \) |
| 61 | \( 1 + (-0.581 - 0.299i)T + (0.580 + 0.814i)T^{2} \) |
| 71 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 73 | \( 1 + (-1.08 - 1.68i)T + (-0.415 + 0.909i)T^{2} \) |
| 79 | \( 1 + (-1.48 + 0.676i)T + (0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (0.580 - 0.814i)T^{2} \) |
| 89 | \( 1 + (-0.786 + 0.618i)T^{2} \) |
| 97 | \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.191925080799422145138658155875, −8.600117992493527701694074292852, −7.950393988558573985976943580207, −6.93754871670347964711559798518, −6.72630929388326329061390660411, −5.18553589512522110013628709973, −4.29205133098479473744874354387, −3.34680931964269427662354881130, −2.35045862259292803587477457114, −0.62332212223731833816210926178,
1.91370092951444069283738025145, 3.13190490375186629880607226617, 4.19860601114925301618355492291, 4.94005777048523631934329449480, 5.75064982365152843346850406575, 6.41088310395574316350054311329, 8.110196722246923343065949569359, 8.449437515699588250505859004124, 9.441080506424418700243560361870, 9.775469495730678785522055241708