L(s) = 1 | − 5.20i·2-s − 19.0·4-s − 24.4i·7-s + 57.4i·8-s − 28.9·11-s − 65.3i·13-s − 126.·14-s + 146.·16-s − 68.1i·17-s − 104.·19-s + 150. i·22-s − 154. i·23-s − 340.·26-s + 464. i·28-s + 205.·29-s + ⋯ |
L(s) = 1 | − 1.83i·2-s − 2.38·4-s − 1.31i·7-s + 2.53i·8-s − 0.794·11-s − 1.39i·13-s − 2.42·14-s + 2.28·16-s − 0.972i·17-s − 1.26·19-s + 1.46i·22-s − 1.40i·23-s − 2.56·26-s + 3.13i·28-s + 1.31·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5939780254\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5939780254\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 5.20iT - 8T^{2} \) |
| 7 | \( 1 + 24.4iT - 343T^{2} \) |
| 11 | \( 1 + 28.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 68.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 154. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 205.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 18.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 337. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 334. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 5.00iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 319. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 430.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 594.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 195. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 929. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 24.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 84.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + 827. iT - 9.12e5T^{2} \) |
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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
−9.923788457087067410563906709865, −8.547819293126021313861241693393, −7.974419144598456965370932301748, −6.62651635333911371211059839358, −5.01968882865059807055323890372, −4.42341650003283301003544025171, −3.25187007387353048281582502364, −2.50360962358946449610175176023, −0.969093584635709347704566490879, −0.20604686625198852874512881354,
2.09991871873655084352031644837, 3.89088142332429302203860439398, 4.92251823871788678271149017096, 5.75450931492486003774020820337, 6.39809765747283885470274749351, 7.30742308634950985235320591166, 8.317180492262126466353327027812, 8.807196798815046496242127271612, 9.526827462673495767294920467716, 10.69082772970058549305236857753