| L(s) = 1 | + 3·3-s − 31·5-s − 54.7i·7-s − 72·9-s + (−11 + 120. i)11-s − 186. i·13-s − 93·15-s − 230. i·17-s − 98.5i·19-s − 164. i·21-s + 277·23-s + 336·25-s − 459·27-s − 1.27e3i·29-s − 1.36e3·31-s + ⋯ |
| L(s) = 1 | + 0.333·3-s − 1.23·5-s − 1.11i·7-s − 0.888·9-s + (−0.0909 + 0.995i)11-s − 1.10i·13-s − 0.413·15-s − 0.795i·17-s − 0.273i·19-s − 0.372i·21-s + 0.523·23-s + 0.537·25-s − 0.629·27-s − 1.51i·29-s − 1.41·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4140669965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4140669965\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (11 - 120. i)T \) |
| good | 3 | \( 1 - 3T + 81T^{2} \) |
| 5 | \( 1 + 31T + 625T^{2} \) |
| 7 | \( 1 + 54.7iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 186. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 230. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 98.5iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 277T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.27e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.36e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 167T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.06e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.20e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.70e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 4.52e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.36e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 3.96e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.80e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.39e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 3.31e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 6.09e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 832. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.67e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 4.24e3T + 8.85e7T^{2} \) |
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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
−10.06853075379051633131810298118, −9.189871428937191474857315233752, −7.995560491095974618971981760777, −7.67701933808390728888563043843, −6.84810277583866490133958792451, −5.42478531134846722144719256251, −4.39515106126636700288914919212, −3.57575103950064562959230794203, −2.58039569036918823535186869760, −0.77856916563066292387795694586,
0.12824625866850383960652776643, 1.89553145880538082450455823708, 3.16457504191411630658085904645, 3.81870389034627952496839244213, 5.18690037220127878714322917626, 6.01716556082621729049306510803, 7.13012360297796934388139903105, 8.175886485379394366901319879178, 8.730528984883917210880206520326, 9.239860900482506636705237142797
