| 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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.447029781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.447029781\) |
| \(L(3)\) |
|
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
−9.701177992853790148407488577224, −8.459268641425128909254708176541, −8.034468085168251611575578943074, −6.59171925546249510149510804511, −6.08599603695016329654162837849, −5.43141940786624349150666698809, −3.83606191625682840830235668712, −2.84696832523874925084751924563, −1.44052210565510214617389327207, −0.871446325413227789793506068191,
1.34171299225696009296017255578, 2.29717314345385644810356009996, 3.32271186262600257901622646808, 4.87945393588888997533330920432, 5.41606783869797412696674435431, 6.25425749295202438799405901440, 7.23729417108824825444295307302, 8.599103610628997642565927978447, 9.304833063160792265982932436839, 9.831538684150230803601101877733
