L(s) = 1 | + (−1.08 + 0.909i)2-s + (−1.32 − 0.483i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (1.87 − 0.684i)6-s + (0.766 + 0.642i)9-s + (1.08 + 0.909i)10-s + (−0.707 + 1.22i)12-s + (1.32 − 0.483i)13-s + (−0.245 + 1.39i)15-s + (0.939 + 0.342i)16-s − 1.41·18-s − 1.00·20-s + (−0.939 + 0.342i)25-s + (−0.999 + 1.73i)26-s + ⋯ |
L(s) = 1 | + (−1.08 + 0.909i)2-s + (−1.32 − 0.483i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (1.87 − 0.684i)6-s + (0.766 + 0.642i)9-s + (1.08 + 0.909i)10-s + (−0.707 + 1.22i)12-s + (1.32 − 0.483i)13-s + (−0.245 + 1.39i)15-s + (0.939 + 0.342i)16-s − 1.41·18-s − 1.00·20-s + (−0.939 + 0.342i)25-s + (−0.999 + 1.73i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3357149794\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3357149794\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (1.08 - 0.909i)T + (0.173 - 0.984i)T^{2} \) |
| 3 | \( 1 + (1.32 + 0.483i)T + (0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.32 + 0.483i)T + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.245 + 1.39i)T + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (1.08 + 0.909i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-1.08 + 0.909i)T + (0.173 - 0.984i)T^{2} \) |
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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
−9.117253184464622146965083949953, −8.402340590831202336568180776200, −7.83839239677226577554220997837, −6.95153683815487225207121391669, −6.14631190221241046026892179166, −5.72703555347456897508434160732, −4.80347374695281852946417682883, −3.59982608745578708669701935652, −1.45888709436266364234402245954, −0.54542252911770121957812014451,
1.16145643419303792214584905934, 2.52610247492520836734286134394, 3.59357063301225749018263706735, 4.55244719803694623852970634402, 5.86880241614770173262660785248, 6.21225525464538832219178082596, 7.31530723543424235123443459576, 8.207416724356486193959675098830, 9.115742684208134934491106542606, 9.834456604049416608099916883478