| L(s) = 1 | − 1.73i·3-s − 0.417·5-s + 2.64i·7-s − 2.99·9-s − 18.4i·11-s − 1.16·13-s + 0.723i·15-s + 0.417·17-s − 21.1i·19-s + 4.58·21-s − 16.9i·23-s − 24.8·25-s + 5.19i·27-s + 4.33·29-s − 20.7i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.0834·5-s + 0.377i·7-s − 0.333·9-s − 1.67i·11-s − 0.0896·13-s + 0.0482i·15-s + 0.0245·17-s − 1.11i·19-s + 0.218·21-s − 0.738i·23-s − 0.993·25-s + 0.192i·27-s + 0.149·29-s − 0.670i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.590451 - 1.02269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.590451 - 1.02269i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 + 0.417T + 25T^{2} \) |
| 11 | \( 1 + 18.4iT - 121T^{2} \) |
| 13 | \( 1 + 1.16T + 169T^{2} \) |
| 17 | \( 1 - 0.417T + 289T^{2} \) |
| 19 | \( 1 + 21.1iT - 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 - 4.33T + 841T^{2} \) |
| 31 | \( 1 + 20.7iT - 961T^{2} \) |
| 37 | \( 1 + 61.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 9.07T + 1.68e3T^{2} \) |
| 43 | \( 1 + 7.30iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 19.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 92.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 99.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 78.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 77.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 43.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 81.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 30.1T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.27890456395864134736382526767, −10.17945465877580053087217901822, −8.848768200980844320217740317506, −8.361779582413651374301447290664, −7.14201773731197661869888113690, −6.15662948415798199815158863989, −5.24001782410416655106249149361, −3.62291412141551214660322493486, −2.37394789608776277719178775859, −0.54052472070413698371622727401,
1.85721004119262984279634473062, 3.57544154818294840941504664044, 4.53854952614387676529738371593, 5.60730307768231683222680155054, 6.97764077593956896849042294250, 7.78757809077339233482127021946, 8.995273771747457484699349709066, 10.01806368367465528307738401839, 10.39014288144427584531024385603, 11.75410280041355829358709522726
