L(s) = 1 | + 2.58i·2-s + 1.31·4-s − 22.8i·7-s + 24.0i·8-s − 11.0·11-s + 11.6i·13-s + 59.1·14-s − 51.7·16-s − 10.0i·17-s − 117.·19-s − 28.6i·22-s + 172. i·23-s − 30.0·26-s − 30.1i·28-s + 178.·29-s + ⋯ |
L(s) = 1 | + 0.913i·2-s + 0.164·4-s − 1.23i·7-s + 1.06i·8-s − 0.303·11-s + 0.248i·13-s + 1.12·14-s − 0.808·16-s − 0.143i·17-s − 1.42·19-s − 0.277i·22-s + 1.56i·23-s − 0.226·26-s − 0.203i·28-s + 1.14·29-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
\(L(2)\) |
\(\approx\) |
\(1.510248586\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.510248586\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.58iT - 8T^{2} \) |
| 7 | \( 1 + 22.8iT - 343T^{2} \) |
| 11 | \( 1 + 11.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 10.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 250. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 600. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 201. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 415.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 531. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 933.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 560. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 538. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 686.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 714. iT - 9.12e5T^{2} \) |
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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
−10.50095460158814244107372076082, −9.606099792199134691164501205148, −8.325060332456036963623732889240, −7.81478741634716911865501916613, −6.83091679118390245357540911994, −6.34290774695983823583624941516, −5.10010187039686748603239837487, −4.20630393739496443605964223859, −2.83540361314453437656566794656, −1.36438653697264200056074535598,
0.40687394452876378417062459362, 2.08339020317243833466498728082, 2.61318364556371737671454586965, 3.86340679150634993406017996379, 5.05814715412777904092416195038, 6.20203177633819710747175330378, 6.89563993422403476088458405858, 8.388350404385570280163792998993, 8.794327034423125118611662588401, 10.15619441029819552799155499410