| L(s) = 1 | + 1.14e3i·3-s − 1.57e4·5-s − 2.88e5i·7-s + 3.47e6·9-s + 5.71e6i·11-s + 2.57e7·13-s − 1.80e7i·15-s − 2.60e8·17-s + 8.37e8i·19-s + 3.28e8·21-s − 5.88e9i·23-s − 5.85e9·25-s + 9.43e9i·27-s + 1.96e10·29-s − 3.89e10i·31-s + ⋯ |
| L(s) = 1 | + 0.522i·3-s − 0.201·5-s − 0.349i·7-s + 0.727·9-s + 0.293i·11-s + 0.409·13-s − 0.105i·15-s − 0.634·17-s + 0.937i·19-s + 0.182·21-s − 1.72i·23-s − 0.959·25-s + 0.901i·27-s + 1.14·29-s − 1.41i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(2.108103053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.108103053\) |
| \(L(8)\) |
|
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 | \( 1 - 1.14e3iT - 4.78e6T^{2} \) |
| 5 | \( 1 + 1.57e4T + 6.10e9T^{2} \) |
| 7 | \( 1 + 2.88e5iT - 6.78e11T^{2} \) |
| 11 | \( 1 - 5.71e6iT - 3.79e14T^{2} \) |
| 13 | \( 1 - 2.57e7T + 3.93e15T^{2} \) |
| 17 | \( 1 + 2.60e8T + 1.68e17T^{2} \) |
| 19 | \( 1 - 8.37e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 5.88e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 - 1.96e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 3.89e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 3.78e10T + 9.01e21T^{2} \) |
| 41 | \( 1 - 5.95e10T + 3.79e22T^{2} \) |
| 43 | \( 1 + 4.00e11iT - 7.38e22T^{2} \) |
| 47 | \( 1 - 4.72e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 5.00e11T + 1.37e24T^{2} \) |
| 59 | \( 1 - 2.44e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 2.89e12T + 9.87e24T^{2} \) |
| 67 | \( 1 - 7.00e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 + 5.99e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 1.03e13T + 1.22e26T^{2} \) |
| 79 | \( 1 - 1.03e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 + 3.46e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 6.54e13T + 1.95e27T^{2} \) |
| 97 | \( 1 - 7.10e13T + 6.52e27T^{2} \) |
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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
−12.02501940843013653015395154639, −10.68366650847273558349185953402, −9.961420712262793291609827659581, −8.654788546030805674235967437458, −7.41471024378108830473574356349, −6.17402522001551176887913553270, −4.56934126979721593148251089596, −3.82564197988602193769880367632, −2.14291747146980117286729679047, −0.66456042119104901415938104724,
0.845343010096820395860189538820, 2.00634345684639968044025373344, 3.47670423929999057381270612386, 4.88640164908882561590963766715, 6.31773097727974055828465850329, 7.34665912278631168629127741064, 8.521661944567474363954816723764, 9.710847567887469116965191943441, 11.07146701663881149027490868995, 12.05310578214201417993313748819
