| L(s) = 1 | − 63.9i·3-s + 217.·5-s − 1.72e3i·7-s + 2.47e3·9-s − 5.95e3i·11-s − 4.91e4·13-s − 1.39e4i·15-s − 5.56e4·17-s − 2.02e5i·19-s − 1.10e5·21-s − 2.09e5i·23-s − 3.43e5·25-s − 5.77e5i·27-s + 3.08e3·29-s + 1.36e6i·31-s + ⋯ |
| L(s) = 1 | − 0.789i·3-s + 0.348·5-s − 0.718i·7-s + 0.377·9-s − 0.406i·11-s − 1.71·13-s − 0.274i·15-s − 0.666·17-s − 1.55i·19-s − 0.567·21-s − 0.748i·23-s − 0.878·25-s − 1.08i·27-s + 0.00435·29-s + 1.47i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.533632 - 1.28830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.533632 - 1.28830i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 63.9iT - 6.56e3T^{2} \) |
| 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
−14.29359822659292338221114305431, −13.30531812509016142195324431582, −12.30111138769884801244957349937, −10.76248687954902712011943446728, −9.400648019126051485386470637534, −7.60628965205222927206051476597, −6.64209756391555686733626621510, −4.65739849250976241896225651397, −2.35812650589682538521024542821, −0.57250205771830771567202340033,
2.19684909296229404503866106389, 4.25647132429671964340172265985, 5.67833114888102104889711452203, 7.57351118866909932846050249272, 9.429371226119558077918527099296, 10.06697530192770405285114237441, 11.75904704586608793154475359205, 12.92932269742977209774518869251, 14.59324131368081621807407851194, 15.35075822714166116392084292964
