| L(s) = 1 | − 0.0979i·2-s + 7.99·4-s − 18.1·5-s + (5.25 + 17.7i)7-s − 1.56i·8-s + 1.77i·10-s + 37.0i·11-s + 18.8i·13-s + (1.73 − 0.514i)14-s + 63.7·16-s − 62.5·17-s − 70.2i·19-s − 144.·20-s + 3.62·22-s − 162. i·23-s + ⋯ |
| L(s) = 1 | − 0.0346i·2-s + 0.998·4-s − 1.62·5-s + (0.283 + 0.958i)7-s − 0.0691i·8-s + 0.0561i·10-s + 1.01i·11-s + 0.403i·13-s + (0.0331 − 0.00982i)14-s + 0.996·16-s − 0.891·17-s − 0.848i·19-s − 1.61·20-s + 0.0351·22-s − 1.47i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.283i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1410212768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1410212768\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-5.25 - 17.7i)T \) |
| good | 2 | \( 1 + 0.0979iT - 8T^{2} \) |
| 5 | \( 1 + 18.1T + 125T^{2} \) |
| 11 | \( 1 - 37.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 18.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 62.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 162. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 95.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 127. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 378.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 158.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 191. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 213.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 220. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 136.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 458. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 967. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 596.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 361.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 35.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.32e3iT - 9.12e5T^{2} \) |
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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
−11.16670383989776700111284806205, −10.11444384403442068152061714892, −8.767149896005221916418086037473, −8.212804591639389005196850375107, −7.05014082115599894534011529914, −6.71272487526722141683301229108, −5.11844963127577270051464727798, −4.21696221123562651782910828650, −2.93840664306515939812162159962, −1.90265952068592902232832545339,
0.04010205535406757435431003620, 1.43728920000253279440116435136, 3.28678025832365156839157132069, 3.77833012531130000014447291485, 5.13521388432348872545202177470, 6.44444468752475321689049527633, 7.31582994151037314401594052720, 7.905724977815522373685263841778, 8.624818960876469267438224546151, 10.25478999118485125432608582522
