| L(s) = 1 | + (168. − 174. i)3-s − 5.34e3i·5-s − 2.81e4·7-s + (−2.09e3 − 5.90e4i)9-s − 1.78e5i·11-s + 4.50e5·13-s + (−9.35e5 − 9.02e5i)15-s − 1.63e6i·17-s − 1.37e6·19-s + (−4.75e6 + 4.92e6i)21-s − 1.19e6i·23-s − 1.88e7·25-s + (−1.06e7 − 9.59e6i)27-s + 7.50e6i·29-s + 3.53e7·31-s + ⋯ |
| L(s) = 1 | + (0.694 − 0.719i)3-s − 1.71i·5-s − 1.67·7-s + (−0.0355 − 0.999i)9-s − 1.10i·11-s + 1.21·13-s + (−1.23 − 1.18i)15-s − 1.14i·17-s − 0.556·19-s + (−1.16 + 1.20i)21-s − 0.186i·23-s − 1.93·25-s + (−0.743 − 0.668i)27-s + 0.365i·29-s + 1.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.608183 + 1.50596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.608183 + 1.50596i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-168. + 174. i)T \) |
| good | 5 | \( 1 + 5.34e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 2.81e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + 1.78e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 4.50e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.63e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 1.37e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 1.19e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 7.50e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 3.53e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 4.95e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 3.90e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 1.40e8T + 2.16e16T^{2} \) |
| 47 | \( 1 - 7.53e4iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 4.45e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 1.38e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 1.60e9T + 7.13e17T^{2} \) |
| 67 | \( 1 - 1.71e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 2.26e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 2.96e8T + 4.29e18T^{2} \) |
| 79 | \( 1 + 5.23e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 7.44e8iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 1.07e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 7.33e9T + 7.37e19T^{2} \) |
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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
−11.64267460340566425080014835594, −9.776860082498040230574758803456, −8.874474372194970849609093018334, −8.324706068996199020224536667983, −6.68729290524170321682208107210, −5.72394251550679579110167738223, −3.96549827426638737151349716157, −2.84386967880744235511239658769, −1.06999269872335806168725925045, −0.41305696361897040865501732936,
2.22492825785537199061567092890, 3.25622032255192562548858556420, 3.98201984163227828073076124451, 6.15769772991206872520973375973, 6.88096697800358616762747899288, 8.289254260857255531507857093636, 9.801098162104840809733734451880, 10.16227397979813991920755722846, 11.19534852029231364850851464550, 12.86467158215022425052050674729
