| L(s) = 1 | + 46.7i·3-s + 722.·5-s + 891. i·7-s − 2.18e3·9-s + 2.57e4i·11-s + 5.89e3·13-s + 3.37e4i·15-s − 1.12e5·17-s − 1.33e5i·19-s − 4.16e4·21-s + 5.38e4i·23-s + 1.31e5·25-s − 1.02e5i·27-s − 1.20e6·29-s + 1.05e6i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.15·5-s + 0.371i·7-s − 0.333·9-s + 1.76i·11-s + 0.206·13-s + 0.667i·15-s − 1.34·17-s − 1.02i·19-s − 0.214·21-s + 0.192i·23-s + 0.335·25-s − 0.192i·27-s − 1.71·29-s + 1.14i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.248624124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248624124\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 46.7iT \) |
| good | 5 | \( 1 - 722.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 891. iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.57e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 5.89e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.12e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.33e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 5.38e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.20e6T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.05e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 7.11e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 4.16e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 1.24e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 4.97e4iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 3.55e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.36e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.96e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 5.61e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 2.39e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.56e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.42e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 6.78e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 5.08e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 6.06e6T + 7.83e15T^{2} \) |
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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
−11.37468348737548137133089907202, −10.44583332915815871974096355269, −9.415843782578136172400083334985, −9.070130870545560981196307584650, −7.37921608065345864525753155706, −6.31007624923791624724889145066, −5.18729125574863221780199669386, −4.29029173495253486742012783008, −2.56633288289462737234422719185, −1.73444148805415396911446512220,
0.25941039468430664822590433132, 1.47392224278542618523373349664, 2.57193428556709280110804601957, 3.97591098529442387737627011484, 5.80136104062904402993680966533, 6.08328427933897479801886936027, 7.48134489963616114888948767707, 8.612740064176555725638624814153, 9.463787917543031374187747255369, 10.73248839884119359167918887461
