| L(s) = 1 | − 1.73i·5-s + 0.990·7-s + 2.10i·11-s + 6.36i·13-s − 3.81·17-s − 2i·19-s − 7.10·23-s + 2.00·25-s − 8.34i·29-s − 2.15·31-s − 1.71i·35-s − 4.62i·37-s − 0.816·41-s − 2.28i·43-s + 6.78·47-s + ⋯ |
| L(s) = 1 | − 0.774i·5-s + 0.374·7-s + 0.633i·11-s + 1.76i·13-s − 0.925·17-s − 0.458i·19-s − 1.48·23-s + 0.400·25-s − 1.54i·29-s − 0.387·31-s − 0.290i·35-s − 0.761i·37-s − 0.127·41-s − 0.348i·43-s + 0.989·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.177350486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177350486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 0.990T + 7T^{2} \) |
| 11 | \( 1 - 2.10iT - 11T^{2} \) |
| 13 | \( 1 - 6.36iT - 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 7.10T + 23T^{2} \) |
| 29 | \( 1 + 8.34iT - 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 + 4.62iT - 37T^{2} \) |
| 41 | \( 1 + 0.816T + 41T^{2} \) |
| 43 | \( 1 + 2.28iT - 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 3.14iT - 53T^{2} \) |
| 59 | \( 1 + 11.9iT - 59T^{2} \) |
| 61 | \( 1 + 4.87iT - 61T^{2} \) |
| 67 | \( 1 + 13.7iT - 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 9.08T + 79T^{2} \) |
| 83 | \( 1 + 4.28iT - 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.059895073306938191443689173923, −7.29856216715785597673251763832, −6.56007959004539825325440291955, −5.90890982290637611388364203114, −4.71141280566438095053196868996, −4.54846411215706935050594596287, −3.71683526946443654512682029612, −2.14810490877487650784198203056, −1.86071904570206749859106116509, −0.32644302790460736632170400537,
1.08636370589864071113434304085, 2.33095571310737458410114255027, 3.11265893063159503727005184498, 3.77871941717672402735588710486, 4.83970457065056837580887226763, 5.61946814842963705847592613029, 6.20803363647151716636923715554, 7.02538537327231094704935123022, 7.74731130441630813718747532637, 8.344675471820503857147045976752
