| L(s) = 1 | − 9.79·3-s − 8i·5-s + 78.3i·7-s + 14.9·9-s + 107.·11-s − 216i·13-s + 78.3i·15-s − 162·17-s − 440.·19-s − 767. i·21-s − 705. i·23-s + 561·25-s + 646.·27-s − 1.30e3i·29-s − 627. i·31-s + ⋯ |
| L(s) = 1 | − 1.08·3-s − 0.320i·5-s + 1.59i·7-s + 0.185·9-s + 0.890·11-s − 1.27i·13-s + 0.348i·15-s − 0.560·17-s − 1.22·19-s − 1.74i·21-s − 1.33i·23-s + 0.897·25-s + 0.887·27-s − 1.55i·29-s − 0.652i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.510482 - 0.510482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.510482 - 0.510482i\) |
| \(L(3)\) |
|
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 | \( 1 + 9.79T + 81T^{2} \) |
| 5 | \( 1 + 8iT - 625T^{2} \) |
| 7 | \( 1 - 78.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 107.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 216iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 162T + 8.35e4T^{2} \) |
| 19 | \( 1 + 440.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 705. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.30e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 627. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.51e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.89e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.90e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.41e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.97e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.26e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 2.37e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.67e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 7.75e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.75e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.99e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 9.33e3T + 4.74e7T^{2} \) |
| 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
−12.33493692089301996841171610895, −11.48275337740360618168539128718, −10.53642352789748138281131459115, −9.102775311454724622978638898187, −8.303002650445643601614777583685, −6.38381933565016609506369513656, −5.78292837073453647806220662514, −4.58293645102061420242842711241, −2.48206631853757054646724900749, −0.37982586459999192134855018660,
1.29696649093180843465772607085, 3.77985337688437496605372513292, 4.88406142502632611400847446449, 6.66415729030919833489966812178, 6.87441255364502176155336622787, 8.713018086859630720791893717218, 10.07691589561799467437696870328, 11.03611390781707402561128709377, 11.53364312666927777070427498948, 12.82495577795500070546908824875
