L(s) = 1 | + (0.707 + 0.707i)2-s + (−1.70 − 0.292i)3-s + 1.00i·4-s + (1.73 + 1.41i)5-s + (−0.999 − 1.41i)6-s + (2.44 − 2.44i)7-s + (−0.707 + 0.707i)8-s + (2.82 + i)9-s + (0.224 + 2.22i)10-s − 5.65i·11-s + (0.292 − 1.70i)12-s + (−3.44 − 3.44i)13-s + 3.46·14-s + (−2.54 − 2.92i)15-s − 1.00·16-s + (5.19 + 5.19i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.985 − 0.169i)3-s + 0.500i·4-s + (0.774 + 0.632i)5-s + (−0.408 − 0.577i)6-s + (0.925 − 0.925i)7-s + (−0.250 + 0.250i)8-s + (0.942 + 0.333i)9-s + (0.0710 + 0.703i)10-s − 1.70i·11-s + (0.0845 − 0.492i)12-s + (−0.956 − 0.956i)13-s + 0.925·14-s + (−0.656 − 0.754i)15-s − 0.250·16-s + (1.26 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78947 + 0.145791i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78947 + 0.145791i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (1.70 + 0.292i)T \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (3.44 + 3.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.19 - 5.19i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + (2.44 - 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 + (-2.89 - 2.89i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.87 - 4.87i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.953 + 0.953i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.51T + 59T^{2} \) |
| 61 | \( 1 - 0.898T + 61T^{2} \) |
| 67 | \( 1 + (10.8 - 10.8i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.6iT - 71T^{2} \) |
| 73 | \( 1 + (1.89 + 1.89i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.44iT - 79T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.12T + 89T^{2} \) |
| 97 | \( 1 + (0.449 - 0.449i)T - 97iT^{2} \) |
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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
−10.59268360437083933242270956404, −10.05607676255817799316206237394, −8.434485212176643862708518520309, −7.63665590752775539144992612222, −6.84066921405699762441714668959, −5.78188458946669073201116173242, −5.44230875316642904253276439154, −4.20859796228025181840564426864, −2.92749906540372938932034199923, −1.08388112414309498813505700366,
1.46267551943397379036372349319, 2.37235773137567227846136848616, 4.43861582749252643503474769174, 4.97283158503024736387380170769, 5.52160287577301073531614414162, 6.69027930406390943032652404582, 7.68002855036089286431920822470, 9.206808109308194305580122545429, 9.723925339034477010709469126027, 10.36676205679819601560152959158