| L(s) = 1 | + 2.82·2-s + 8.00·4-s + 5.79i·5-s + 22.6·8-s + 16.3i·10-s − 10.0·11-s − 190. i·13-s + 64.0·16-s − 421. i·17-s + 432. i·19-s + 46.3i·20-s − 28.3·22-s − 921.·23-s + 591.·25-s − 538. i·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.231i·5-s + 0.353·8-s + 0.163i·10-s − 0.0828·11-s − 1.12i·13-s + 0.250·16-s − 1.45i·17-s + 1.19i·19-s + 0.115i·20-s − 0.0586·22-s − 1.74·23-s + 0.946·25-s − 0.795i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.291732775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291732775\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5.79iT - 625T^{2} \) |
| 11 | \( 1 + 10.0T + 1.46e4T^{2} \) |
| 13 | \( 1 + 190. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 421. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 432. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 921.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 877.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 724. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 540.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 894. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.24e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.75e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 812.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 2.85e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.83e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.20e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.40e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.39e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.35e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 3.75e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 6.36e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 6.37e3iT - 8.85e7T^{2} \) |
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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
−9.380294656020290332285191934199, −8.139176378715087941698387958365, −7.53200502441806767022999447677, −6.52261399559432525997395542633, −5.63356285209825316255541369620, −4.89906136807697180829591196361, −3.68769144337926929494380187550, −2.91996399180446479787533403947, −1.71730700285342166648513451371, −0.19834127206372104568255574462,
1.49096884732988222620748878222, 2.44564049374025048172006511601, 3.81781237986404836218285408034, 4.42184526181481728661969053155, 5.52205768244070662906492135783, 6.35760818632232266353172031271, 7.15193686157710366248544006319, 8.202001565467062805731584753299, 9.000538462791225380703455684182, 9.980981065232578663138477940515
