L(s) = 1 | + (1 − 2i)5-s + 4i·7-s − 4·11-s − 4i·17-s + 4i·23-s + (−3 − 4i)25-s + 6·29-s + 4·31-s + (8 + 4i)35-s + 8i·37-s + 10·41-s + 4i·43-s + 4i·47-s − 9·49-s + 12i·53-s + ⋯ |
L(s) = 1 | + (0.447 − 0.894i)5-s + 1.51i·7-s − 1.20·11-s − 0.970i·17-s + 0.834i·23-s + (−0.600 − 0.800i)25-s + 1.11·29-s + 0.718·31-s + (1.35 + 0.676i)35-s + 1.31i·37-s + 1.56·41-s + 0.609i·43-s + 0.583i·47-s − 1.28·49-s + 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.554183671\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.554183671\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.975964025100136206621664613492, −8.195899155635972310779879105011, −7.66199205490648303859296252261, −6.38817983545297678394646740654, −5.72798517486197668433087073447, −5.09908016536640263023701841801, −4.51479370468263479328035015029, −2.88183198193585032003604208993, −2.46351656050060685852924316645, −1.13630905254918138951108820358,
0.53434555950747201256705787350, 1.99396419588046229985214049995, 2.93200345946838934148684090242, 3.84865174102725762780628519047, 4.63509381443500709508140664893, 5.67699219058747295802009781864, 6.47634634599677172449226370459, 7.11790515786410820470321984898, 7.80236829973923599195956978982, 8.444704966249952465440110312309