L(s) = 1 | + (−1.66 + 0.468i)3-s + 0.762i·5-s − 0.762i·7-s + (2.56 − 1.56i)9-s − 1.78·11-s + 0.799·13-s + (−0.357 − 1.27i)15-s + 4.92i·17-s − 7.11i·19-s + (0.357 + 1.27i)21-s + 23-s + 4.41·25-s + (−3.53 + 3.80i)27-s − 9.21i·29-s + 7.02i·31-s + ⋯ |
L(s) = 1 | + (−0.962 + 0.270i)3-s + 0.340i·5-s − 0.288i·7-s + (0.853 − 0.521i)9-s − 0.538·11-s + 0.221·13-s + (−0.0922 − 0.328i)15-s + 1.19i·17-s − 1.63i·19-s + (0.0779 + 0.277i)21-s + 0.208·23-s + 0.883·25-s + (−0.680 + 0.732i)27-s − 1.71i·29-s + 1.26i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.112780469\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.112780469\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.66 - 0.468i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 0.762iT - 5T^{2} \) |
| 7 | \( 1 + 0.762iT - 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 - 0.799T + 13T^{2} \) |
| 17 | \( 1 - 4.92iT - 17T^{2} \) |
| 19 | \( 1 + 7.11iT - 19T^{2} \) |
| 29 | \( 1 + 9.21iT - 29T^{2} \) |
| 31 | \( 1 - 7.02iT - 31T^{2} \) |
| 37 | \( 1 - 3.78T + 37T^{2} \) |
| 41 | \( 1 - 5.00iT - 41T^{2} \) |
| 43 | \( 1 + 2.95iT - 43T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 - 7.11iT - 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 - 10.8iT - 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 9.78T + 73T^{2} \) |
| 79 | \( 1 - 5.58iT - 79T^{2} \) |
| 83 | \( 1 - 7.38T + 83T^{2} \) |
| 89 | \( 1 + 4.47iT - 89T^{2} \) |
| 97 | \( 1 - 5.38T + 97T^{2} \) |
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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
−10.13397013297789676608633540567, −9.191848674140099932864055186611, −8.226922446503847869653161330830, −7.14841869866582361902399057953, −6.53868701128344329178179354242, −5.63382040025012463833449087628, −4.73065180476101966381487123432, −3.87462323503511660961200789928, −2.56093560685562495621637352999, −0.860081189414698304408183417684,
0.848763617561153517973909552204, 2.23244874665302969444627516958, 3.69027204370765228226535984499, 4.93682425185476662970262730827, 5.44590956740159932884172006661, 6.37267139261746674795317015978, 7.28572110040403944149011240433, 8.034791753641033537970978863134, 9.051652525389621699045843976505, 9.923592799689193613203318181940