| L(s) = 1 | + 15.5·3-s − 35.0i·5-s + 346. i·7-s + 243·9-s + 1.44e3·11-s − 1.40e3i·13-s − 546. i·15-s − 1.26e3·17-s − 3.78e3·19-s + 5.39e3i·21-s + 1.60e4i·23-s + 1.43e4·25-s + 3.78e3·27-s + 1.59e4i·29-s − 3.20e3i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.280i·5-s + 1.00i·7-s + 0.333·9-s + 1.08·11-s − 0.638i·13-s − 0.161i·15-s − 0.257·17-s − 0.551·19-s + 0.582i·21-s + 1.31i·23-s + 0.921·25-s + 0.192·27-s + 0.652i·29-s − 0.107i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.637274599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.637274599\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 15.5T \) |
| good | 5 | \( 1 + 35.0iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 346. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.44e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.40e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 1.26e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 3.78e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.60e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.59e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.20e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.55e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 4.15e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.39e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.48e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.55e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.83e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 3.63e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.53e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 4.83e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.48e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.53e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 9.93e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 5.90e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 9.64e5T + 8.32e11T^{2} \) |
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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
−11.73718914645682093555921988044, −10.53654057626963622358088057806, −9.201735771122288852681315932069, −8.836255340834551580237377577805, −7.61818183337016082583863019236, −6.35461960579376427199415387590, −5.19948091327397535221650171283, −3.82124937299159880514908898217, −2.56205218473834792876090405436, −1.21902781520549728553392802112,
0.76573705955812784769555249462, 2.20857818210649479799474928851, 3.71817570050606621385023422429, 4.52265385790670574571364345595, 6.45306792325988111475020669699, 7.09200039967812228869630219924, 8.369401155362594875289369057379, 9.278965840951944902242411275728, 10.34481782869901820441901398342, 11.19208512975657541801375234499
