| 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
−8.344675471820503857147045976752, −7.74731130441630813718747532637, −7.02538537327231094704935123022, −6.20803363647151716636923715554, −5.61946814842963705847592613029, −4.83970457065056837580887226763, −3.77871941717672402735588710486, −3.11265893063159503727005184498, −2.33095571310737458410114255027, −1.08636370589864071113434304085,
0.32644302790460736632170400537, 1.86071904570206749859106116509, 2.14810490877487650784198203056, 3.71683526946443654512682029612, 4.54846411215706935050594596287, 4.71141280566438095053196868996, 5.90890982290637611388364203114, 6.56007959004539825325440291955, 7.29856216715785597673251763832, 8.059895073306938191443689173923
