| L(s) = 1 | + 1.14e3i·5-s + 618.·7-s + 5.40e3i·11-s + 2.45e4·13-s − 1.13e4i·17-s − 3.87e4·19-s − 3.47e5i·23-s − 9.12e5·25-s − 5.13e5i·29-s + 1.02e6·31-s + 7.05e5i·35-s − 2.38e6·37-s + 4.26e6i·41-s − 3.81e6·43-s − 3.71e6i·47-s + ⋯ |
| L(s) = 1 | + 1.82i·5-s + 0.257·7-s + 0.369i·11-s + 0.859·13-s − 0.135i·17-s − 0.297·19-s − 1.24i·23-s − 2.33·25-s − 0.725i·29-s + 1.11·31-s + 0.470i·35-s − 1.27·37-s + 1.50i·41-s − 1.11·43-s − 0.760i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.4935400271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4935400271\) |
| \(L(5)\) |
|
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 | \( 1 - 1.14e3iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 618.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 5.40e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.45e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.13e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 3.87e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.47e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 5.13e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.02e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.38e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 4.26e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.81e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.71e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 6.10e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.02e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.28e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.05e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 7.37e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.85e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.10e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.33e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.49e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 2.06e7T + 7.83e15T^{2} \) |
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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
−9.928467372087188220707107736080, −8.547919249602131378876390307031, −7.69504811319682026942425323698, −6.60855527708035782669055342212, −6.27193926879709305033742308304, −4.74769715513142463879594780780, −3.57763793348940339813923537337, −2.73647649748508919686605202116, −1.72696081184901965926157209338, −0.092625370636652013999604552437,
1.08155261834890182808564752725, 1.68361779633952009715377841994, 3.41811148265853697463198894302, 4.43984228106724144567601983547, 5.27859031577542335381880520990, 6.08030924978726209424461532486, 7.50599102413842064423008062362, 8.554442383647271424366019974153, 8.825079863241881498488437580618, 9.895505205414489986764756834962
