| L(s) = 1 | + 12i·5-s + 5.19i·7-s + 62.3·11-s + 7·13-s − 84i·17-s + 67.5i·19-s − 62.3·23-s − 19·25-s + 168i·29-s + 259. i·31-s − 62.3·35-s − 97·37-s + 72i·41-s + 363. i·43-s − 436.·47-s + ⋯ |
| L(s) = 1 | + 1.07i·5-s + 0.280i·7-s + 1.70·11-s + 0.149·13-s − 1.19i·17-s + 0.815i·19-s − 0.565·23-s − 0.151·25-s + 1.07i·29-s + 1.50i·31-s − 0.301·35-s − 0.430·37-s + 0.274i·41-s + 1.28i·43-s − 1.35·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.902853602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902853602\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 12iT - 125T^{2} \) |
| 7 | \( 1 - 5.19iT - 343T^{2} \) |
| 11 | \( 1 - 62.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 67.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 62.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 168iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 259. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 97T + 5.06e4T^{2} \) |
| 41 | \( 1 - 72iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 363. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 504iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 436.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 133T + 2.26e5T^{2} \) |
| 67 | \( 1 - 545. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 498.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 497T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 872.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.16e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 749T + 9.12e5T^{2} \) |
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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
−11.04869271204139770479744137674, −10.04173395899641769414687212163, −9.244428523339295185568538494241, −8.261918711209258603287937108009, −6.91693747512994848825464126872, −6.58583534537462460750784443565, −5.27446917888614101625513221848, −3.87143365449609969800654649629, −2.93890193350997548980558054556, −1.43091112574679310413471752487,
0.67642282922318683444706751879, 1.85327749726469548426947438164, 3.80309465254805331243007853393, 4.45688962217178384699447685032, 5.79588771665781368154260526421, 6.65703171997621983240572098479, 7.893400882531479888949978768263, 8.825419251964789224374093174785, 9.391823153486780834160672070260, 10.48034791464414079501552583120
