| L(s) = 1 | + 5.19i·3-s − 39.7·5-s − 46.0i·7-s − 27·9-s + 181. i·11-s − 183.·13-s − 206. i·15-s + 427.·17-s − 668. i·19-s + 239.·21-s − 882. i·23-s + 954.·25-s − 140. i·27-s − 807.·29-s − 391. i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.58·5-s − 0.939i·7-s − 0.333·9-s + 1.49i·11-s − 1.08·13-s − 0.917i·15-s + 1.48·17-s − 1.85i·19-s + 0.542·21-s − 1.66i·23-s + 1.52·25-s − 0.192i·27-s − 0.959·29-s − 0.407i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6635227695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6635227695\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.19iT \) |
| good | 5 | \( 1 + 39.7T + 625T^{2} \) |
| 7 | \( 1 + 46.0iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 181. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 183.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 427.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 668. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 882. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 807.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 391. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 466.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.15e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 509. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 753.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.30e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 801.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 505. iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.17e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.29e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.07e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 2.97e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.49e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 8.18e3T + 8.85e7T^{2} \) |
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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
−10.06624127764499248250585973188, −9.201163957970456961441174317233, −8.067105395671234384302415332016, −7.33381298528621546104792938564, −6.91102062257333395608421763179, −5.02173309814050758394465762178, −4.50324497334772381965886309264, −3.72815668390396722875002593965, −2.57052382315564330932474597316, −0.67663127283522962466601035877,
0.24714341642059371992173833528, 1.59953417915793974901434503938, 3.23062639887261087333972270436, 3.61449410902695006894465402974, 5.29850951394329622486168465520, 5.83173724546239487538726294409, 7.19829215109939957947379944942, 7.925611436609507359047849870452, 8.315578492415147603094220721127, 9.391388166813394629933044940459
