| L(s) = 1 | + 435. i·5-s − 34.1·7-s + 3.88e3i·11-s − 2.56e3i·13-s + 2.32e4·17-s − 2.38e4i·19-s − 6.04e4·23-s − 1.11e5·25-s − 3.93e4i·29-s − 7.04e4·31-s − 1.48e4i·35-s + 2.93e5i·37-s − 3.77e5·41-s + 1.01e6i·43-s + 2.19e5·47-s + ⋯ |
| L(s) = 1 | + 1.55i·5-s − 0.0376·7-s + 0.879i·11-s − 0.323i·13-s + 1.14·17-s − 0.797i·19-s − 1.03·23-s − 1.43·25-s − 0.299i·29-s − 0.424·31-s − 0.0586i·35-s + 0.954i·37-s − 0.855·41-s + 1.95i·43-s + 0.308·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.05474440144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05474440144\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 435. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 34.1T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.88e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 2.56e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.38e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 6.04e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.93e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 7.04e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.93e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.01e6iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 2.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.74e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.40e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.75e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.00e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.06e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.17e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 3.89e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.87e6T + 8.07e13T^{2} \) |
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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
−9.827689026708452889554945140567, −8.253188044494481725970057358501, −7.44938022497856911238042974345, −6.72906186240479914107262903661, −5.88313251121319706998782013263, −4.67137669235170673607711975924, −3.45037274438999891398822731570, −2.72705855728268404904116596905, −1.62242120913369670031461205016, −0.01077099369620408158576725613,
0.981416756164718725134344947955, 1.84441984677747658607421219456, 3.43320209143751308813407117148, 4.28965145226464887971017486002, 5.43759555286042420926168409528, 5.89458537532686087055822316341, 7.37797181993526185716293726175, 8.292033897736729514401547287746, 8.845142615071568624462643439989, 9.736700828949141378818274104489
