| L(s) = 1 | + (−8 + 13.8i)2-s + (−127. − 221. i)4-s + (−897. − 1.55e3i)5-s + (5.59e3 − 9.68e3i)7-s + 4.09e3·8-s + 2.87e4·10-s + (−9.21e3 + 1.59e4i)11-s + (−1.05e4 − 1.83e4i)13-s + (8.94e4 + 1.54e5i)14-s + (−3.27e4 + 5.67e4i)16-s − 6.60e5·17-s + 5.22e5·19-s + (−2.29e5 + 3.98e5i)20-s + (−1.47e5 − 2.55e5i)22-s + (5.24e5 + 9.08e5i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.642 − 1.11i)5-s + (0.880 − 1.52i)7-s + 0.353·8-s + 0.908·10-s + (−0.189 + 0.328i)11-s + (−0.102 − 0.178i)13-s + (0.622 + 1.07i)14-s + (−0.125 + 0.216i)16-s − 1.91·17-s + 0.919·19-s + (−0.321 + 0.556i)20-s + (−0.134 − 0.232i)22-s + (0.390 + 0.676i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0594262 - 0.512620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0594262 - 0.512620i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8 - 13.8i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (897. + 1.55e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-5.59e3 + 9.68e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (9.21e3 - 1.59e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (1.05e4 + 1.83e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + 6.60e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.22e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-5.24e5 - 9.08e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-1.30e6 + 2.25e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-2.69e4 - 4.67e4i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + 1.93e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (6.48e5 + 1.12e6i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-7.35e6 + 1.27e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (1.53e7 - 2.65e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + 5.86e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-1.14e7 - 1.98e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (9.03e7 - 1.56e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.18e8 - 2.06e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 4.73e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.05e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-1.28e8 + 2.23e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.46e8 + 2.54e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + 1.41e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (1.64e8 - 2.84e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
| 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
−13.14098757221857254199974854604, −11.60961358293807651196995460750, −10.48667913496140485353069599318, −9.009691543042113938552971244392, −7.932490510268443989573476776467, −7.05186816215606579676043617937, −4.99726477221667656183999142376, −4.19369761004185914785400843636, −1.35115524022040936257773494790, −0.19597336470791331288975945485,
2.03705416386395345949931130452, 3.12131542964368713197980027635, 4.92262108475299313413712004792, 6.72733924494185402428078328247, 8.183503283282396083999763779205, 9.129145424935085599665684164709, 10.85610339528578889073519625041, 11.39887650471582472163017040599, 12.41480997672745238566966306703, 14.00384448887038030670083041569
