L(s) = 1 | + 2.30i·2-s − 3.30·4-s + 2.60i·7-s − 3.00i·8-s + 4.60·11-s + 6.60i·13-s − 6·14-s + 0.302·16-s − 1.60i·17-s − 3.60·19-s + 10.6i·22-s − 3i·23-s − 15.2·26-s − 8.60i·28-s − 1.39·29-s + ⋯ |
L(s) = 1 | + 1.62i·2-s − 1.65·4-s + 0.984i·7-s − 1.06i·8-s + 1.38·11-s + 1.83i·13-s − 1.60·14-s + 0.0756·16-s − 0.389i·17-s − 0.827·19-s + 2.26i·22-s − 0.625i·23-s − 2.98·26-s − 1.62i·28-s − 0.258·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.282551 - 1.19690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.282551 - 1.19690i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2.30iT - 2T^{2} \) |
| 7 | \( 1 - 2.60iT - 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 6.60iT - 13T^{2} \) |
| 17 | \( 1 + 1.60iT - 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 4.60T + 41T^{2} \) |
| 43 | \( 1 - 0.605iT - 43T^{2} \) |
| 47 | \( 1 - 9.21iT - 47T^{2} \) |
| 53 | \( 1 - 1.60iT - 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 - 0.788iT - 67T^{2} \) |
| 71 | \( 1 - 7.39T + 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 3iT - 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 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
−11.12828188872104221444779500042, −9.466039048913983155427190265871, −9.066079287695574933004183147155, −8.454508847258919976208723225796, −7.21878542517889778180208383321, −6.55561364681611844453535830232, −5.96113950496678308390715207903, −4.80068737351865729137662196521, −3.98527890432724158685654259874, −2.04575594871964318394423956309,
0.67692972901026458455109863076, 1.86710467359163906338031227725, 3.42031498201743255397883523954, 3.84459755256625864013118812654, 5.05903544583601989574463104512, 6.39167326965225354527598354564, 7.52490253636756123154242469781, 8.596380389409863398849716600832, 9.459321573542760774861707261327, 10.37604016819879575772007519633