| L(s) = 1 | + 0.732i·3-s − 2.73·5-s + (2 + 1.73i)7-s + 2.46·9-s + 1.46·11-s − 1.26·13-s − 2i·15-s + 4i·17-s − 4.73i·19-s + (−1.26 + 1.46i)21-s + 1.46i·23-s + 2.46·25-s + 4i·27-s + 6.92i·29-s − 6.92·31-s + ⋯ |
| L(s) = 1 | + 0.422i·3-s − 1.22·5-s + (0.755 + 0.654i)7-s + 0.821·9-s + 0.441·11-s − 0.351·13-s − 0.516i·15-s + 0.970i·17-s − 1.08i·19-s + (−0.276 + 0.319i)21-s + 0.305i·23-s + 0.492·25-s + 0.769i·27-s + 1.28i·29-s − 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0716 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0716 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.865450 + 0.929814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.865450 + 0.929814i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| good | 3 | \( 1 - 0.732iT - 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 4.73iT - 19T^{2} \) |
| 23 | \( 1 - 1.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 9.46T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 6.92iT - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 + 5.66T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 7.26iT - 83T^{2} \) |
| 89 | \( 1 - 1.07iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−10.57202342774936466804578744249, −9.272576273084422761073202230541, −8.791857672560563263726061455027, −7.69571184833196395164769811316, −7.22012803478872992568963484150, −5.93077526664128178923172511820, −4.72819708528956310774108383687, −4.23032418459533185595911699425, −3.10497425902761634594239660906, −1.50965111345812625728243321209,
0.66277459923089049582352058904, 2.07947298246006322917760616113, 3.84777846204173715390377319389, 4.21856786524656515731551904812, 5.42861085727568203242823398289, 6.78954321362138876887089126078, 7.57944556271116568320813633307, 7.79183636075854223008215260015, 8.967695175876395147018565649484, 9.972251199062046823164135418038
