| L(s) = 1 | + 46.7i·3-s + 440. i·5-s + (−2.39e3 + 166. i)7-s − 2.18e3·9-s + 1.47e4·11-s + 2.84e3i·13-s − 2.06e4·15-s − 4.42e4i·17-s + 7.04e4i·19-s + (−7.79e3 − 1.12e5i)21-s + 1.95e5·23-s + 1.96e5·25-s − 1.02e5i·27-s − 1.35e5·29-s − 5.68e5i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.705i·5-s + (−0.997 + 0.0693i)7-s − 0.333·9-s + 1.00·11-s + 0.0994i·13-s − 0.407·15-s − 0.530i·17-s + 0.540i·19-s + (−0.0400 − 0.575i)21-s + 0.700·23-s + 0.502·25-s − 0.192i·27-s − 0.191·29-s − 0.615i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0693 - 0.997i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0693 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.960348930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.960348930\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 46.7iT \) |
| 7 | \( 1 + (2.39e3 - 166. i)T \) |
| good | 5 | \( 1 - 440. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 1.47e4T + 2.14e8T^{2} \) |
| 13 | \( 1 - 2.84e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 4.42e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 7.04e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.95e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 1.35e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 5.68e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.04e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.65e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.62e5T + 1.16e13T^{2} \) |
| 47 | \( 1 - 2.45e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.90e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.47e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.62e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 2.53e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 8.36e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + 4.40e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 1.01e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 6.64e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.88e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.00e8iT - 7.83e15T^{2} \) |
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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
−10.34115014987206058810257724366, −9.509195240842908775048175859552, −8.868718556462385004732672504807, −7.41203096773244100950191635605, −6.56362488930474784946345162023, −5.71732646990178799822739878594, −4.30249591002069320035844570461, −3.40872060384701063449000943089, −2.49776173875633009197354996849, −0.827502539010846786763676621615,
0.54153358758240086791089106254, 1.35389298043493532906485952266, 2.76307721847099747396791283781, 3.87740024336737668383564276476, 5.07916324322358249944407950703, 6.30265125259589132008854918273, 6.88033866494692289232365832727, 8.126499005808470349007589748140, 9.039520959047478819689454464695, 9.694063377161719490977465516013
