| L(s) = 1 | + 9.79i·3-s − 8·5-s + 78.3i·7-s − 14.9·9-s + 107. i·11-s + 216·13-s − 78.3i·15-s − 162·17-s + 440. i·19-s − 767.·21-s − 705. i·23-s − 561·25-s + 646. i·27-s + 1.30e3·29-s + 627. i·31-s + ⋯ |
| L(s) = 1 | + 1.08i·3-s − 0.320·5-s + 1.59i·7-s − 0.185·9-s + 0.890i·11-s + 1.27·13-s − 0.348i·15-s − 0.560·17-s + 1.22i·19-s − 1.74·21-s − 1.33i·23-s − 0.897·25-s + 0.887i·27-s + 1.55·29-s + 0.652i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.500916066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500916066\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 9.79iT - 81T^{2} \) |
| 5 | \( 1 + 8T + 625T^{2} \) |
| 7 | \( 1 - 78.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 107. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 216T + 2.85e4T^{2} \) |
| 17 | \( 1 + 162T + 8.35e4T^{2} \) |
| 19 | \( 1 - 440. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 705. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.30e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 627. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.51e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.89e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.90e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.41e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.97e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.26e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.37e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.67e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 7.75e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.75e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 7.99e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.33e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.43e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 7.45e3T + 8.85e7T^{2} \) |
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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
−11.99502048964561741168399576424, −10.67148846493055941537919134196, −10.02760092834361899509521329772, −8.850018481361491785686353462638, −8.404139545623759642026174262220, −6.70294202830866211167798529811, −5.58469553594060327510550806037, −4.54936734857649346757485627329, −3.47967192803945808313665668399, −1.96687028265779541582761247550,
0.51594126251091815546099918638, 1.42203651582824145975922378515, 3.32047374592126350215691885422, 4.41765386654796019949582291286, 6.14923762325905615251301019274, 6.96360960757885403079376117367, 7.76435938634037035132602302670, 8.656890845022199942237746518370, 10.08420758803752556298567877855, 11.11257689308821343968181328677
