| L(s) = 1 | − 217.·5-s + 1.72e3i·7-s − 5.95e3i·11-s − 4.91e4·13-s + 5.56e4·17-s + 2.02e5i·19-s − 2.09e5i·23-s − 3.43e5·25-s − 3.08e3·29-s − 1.36e6i·31-s − 3.75e5i·35-s + 1.31e6·37-s − 4.54e6·41-s + 3.48e6i·43-s + 7.23e6i·47-s + ⋯ |
| L(s) = 1 | − 0.348·5-s + 0.718i·7-s − 0.406i·11-s − 1.71·13-s + 0.666·17-s + 1.55i·19-s − 0.748i·23-s − 0.878·25-s − 0.00435·29-s − 1.47i·31-s − 0.250i·35-s + 0.701·37-s − 1.60·41-s + 1.02i·43-s + 1.48i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.141282040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.141282040\) |
| \(L(5)\) |
|
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 | \( 1 + 217.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 1.72e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 5.95e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.91e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.56e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 2.02e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 2.09e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.08e3T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.36e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.31e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 4.54e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 3.48e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 7.23e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.48e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.48e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 4.13e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 6.04e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 4.20e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 9.00e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 6.81e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 5.81e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.82e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 6.01e7T + 7.83e15T^{2} \) |
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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
−10.09047675967155330123172257079, −9.512134598148815646928313903522, −8.180801038007704519498104430026, −7.63451493535909157179846858521, −6.25077015172947334576103895374, −5.36676015915262584933321223302, −4.20980981612918985346027768168, −2.96313896395331317369780066053, −1.90766330975235832958936443624, −0.33844341703226508853590705427,
0.69459831318461895127260168633, 2.13362166571547993771128887386, 3.35812249711331330950907291933, 4.54054769389466582498012500415, 5.36751609125962511437993008076, 7.10948207956894547551520112571, 7.28376067988661618941714309153, 8.612010512257175975949356103661, 9.764287225469972230181453833071, 10.35013075292088605971122963136
