| L(s) = 1 | − 374. i·3-s − 5.15e3·5-s − 2.04e4i·7-s − 8.12e4·9-s + 2.70e5i·11-s − 1.59e5·13-s + 1.92e6i·15-s − 2.36e6·17-s − 3.01e5i·19-s − 7.65e6·21-s + 5.09e5i·23-s + 1.67e7·25-s + 8.30e6i·27-s + 1.59e6·29-s + 2.19e7i·31-s + ⋯ |
| L(s) = 1 | − 1.54i·3-s − 1.64·5-s − 1.21i·7-s − 1.37·9-s + 1.68i·11-s − 0.429·13-s + 2.54i·15-s − 1.66·17-s − 0.121i·19-s − 1.87·21-s + 0.0791i·23-s + 1.71·25-s + 0.578i·27-s + 0.0775·29-s + 0.765i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.6194456882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6194456882\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 374. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 5.15e3T + 9.76e6T^{2} \) |
| 7 | \( 1 + 2.04e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 2.70e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 1.59e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.36e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 3.01e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 5.09e5iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 1.59e6T + 4.20e14T^{2} \) |
| 31 | \( 1 - 2.19e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 6.44e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 9.46e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 1.15e7iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 1.22e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 3.36e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 5.71e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 1.47e9T + 7.13e17T^{2} \) |
| 67 | \( 1 - 8.73e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 2.55e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.99e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 1.87e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 5.53e8iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 2.47e8T + 3.11e19T^{2} \) |
| 97 | \( 1 - 1.05e10T + 7.37e19T^{2} \) |
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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
−10.15868930910022860317942324968, −8.642303010305697836264896882415, −7.69805620178197855051361118078, −7.11018030269793455223074406186, −6.77196609297339914323463139755, −4.69488711043311706953857200405, −4.01505358578452897415367364314, −2.52316596166491059454605288542, −1.39238936008326320508183915804, −0.34584081643471417599297740984,
0.34276661838187230559024925159, 2.63236649211819743236084660082, 3.51657308709716303758471171483, 4.32810383034020003525793655679, 5.22065899050241957784574928801, 6.37101306272222919800046570740, 7.997162317155557805791899813032, 8.690666667666099139138069825506, 9.323763405384072031679663488474, 10.72218801557149384690686168905
