| L(s) = 1 | + (8 − 13.8i)2-s + (−3 − 5.19i)3-s + (−127. − 221. i)4-s + (280 − 484. i)5-s − 96·6-s − 4.09e3·8-s + (9.82e3 − 1.70e4i)9-s + (−4.47e3 − 7.75e3i)10-s + (2.70e4 + 4.68e4i)11-s + (−768. + 1.33e3i)12-s + 1.13e5·13-s − 3.36e3·15-s + (−3.27e4 + 5.67e4i)16-s + (3.13e3 + 5.42e3i)17-s + (−1.57e5 − 2.72e5i)18-s + (1.28e5 − 2.22e5i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.0213 − 0.0370i)3-s + (−0.249 − 0.433i)4-s + (0.200 − 0.347i)5-s − 0.0302·6-s − 0.353·8-s + (0.499 − 0.864i)9-s + (−0.141 − 0.245i)10-s + (0.557 + 0.965i)11-s + (−0.0106 + 0.0185i)12-s + 1.09·13-s − 0.0171·15-s + (−0.125 + 0.216i)16-s + (0.00909 + 0.0157i)17-s + (−0.352 − 0.611i)18-s + (0.226 − 0.391i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.69698 - 2.23065i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69698 - 2.23065i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 + 13.8i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (3 + 5.19i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-280 + 484. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-2.70e4 - 4.68e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.13e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-3.13e3 - 5.42e3i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-1.28e5 + 2.22e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.33e5 + 2.30e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 1.57e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (2.31e6 + 4.01e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-5.97e6 + 1.03e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + 2.19e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.75e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.64e7 + 4.58e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (8.11e6 + 1.40e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (7.02e7 + 1.21e8i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.01e8 - 1.75e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (7.68e7 + 1.33e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.79e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (2.02e8 + 3.49e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-6.53e7 + 1.13e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 4.20e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.34e8 - 4.06e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 8.72e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95214811335793928032129252900, −10.83357129122771176205132868319, −9.615747192724575143436813226792, −8.892249268233763691779811404672, −7.14054095748414189535037043277, −5.96124378202503181516345761046, −4.53809755938214586134319490728, −3.50659656983412659066913517309, −1.81284687976489637622760287894, −0.76257220081181064266713950327,
1.24826624450371507782370930583, 3.04332746415443094223628027236, 4.32422307772218678009783588745, 5.71147121032160896762566811941, 6.65096300654346586011837829712, 7.929656467312993824015607817053, 8.917197362096695763340283393851, 10.38051857060496762497354333210, 11.31011507995322492547217654139, 12.64136787460267938547748565225
