| L(s) = 1 | + (−1 + 1.73i)2-s + (14 + 24.2i)4-s + (5.5 − 9.52i)5-s + (129.5 + 6.06i)7-s − 120·8-s + (11 + 19.0i)10-s + (134.5 + 232. i)11-s − 308·13-s + (−140 + 218. i)14-s + (−328 + 568. i)16-s + (948 + 1.64e3i)17-s + (82 − 142. i)19-s + 308·20-s − 538·22-s + (−1.63e3 + 2.82e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.176 + 0.306i)2-s + (0.437 + 0.757i)4-s + (0.0983 − 0.170i)5-s + (0.998 + 0.0467i)7-s − 0.662·8-s + (0.0347 + 0.0602i)10-s + (0.335 + 0.580i)11-s − 0.505·13-s + (−0.190 + 0.297i)14-s + (−0.320 + 0.554i)16-s + (0.795 + 1.37i)17-s + (0.0521 − 0.0902i)19-s + 0.172·20-s − 0.236·22-s + (−0.643 + 1.11i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.25803 + 1.27908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25803 + 1.27908i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-129.5 - 6.06i)T \) |
| good | 2 | \( 1 + (1 - 1.73i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-5.5 + 9.52i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-134.5 - 232. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 308T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-948 - 1.64e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-82 + 142. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.63e3 - 2.82e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 2.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (1.42e3 + 2.46e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-5.66e3 + 9.81e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.68e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.89e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.05e4 + 1.82e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.48e4 + 2.57e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (4.08e3 + 7.06e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (7.58e3 - 1.31e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.60e4 - 2.77e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.74e4 + 3.01e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (6.76e3 - 1.17e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 6.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (5.74e4 - 9.95e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.54e5T + 8.58e9T^{2} \) |
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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
−14.54348893151507141293156939540, −12.95214906774863978430227718158, −12.03362765745625803400697044795, −11.01689256090200368971508731506, −9.434732126130794881203252510319, −8.117822211397690101190197204966, −7.29647728617382301645682747042, −5.64254554922747047630528466922, −3.89150614055319611921541105177, −1.89117142083973604670015071769,
0.940870785067477514325087956900, 2.60172269683925419811545073202, 4.86049214099348166130608297492, 6.22089359202062142048700507918, 7.69358835229540893068209075687, 9.198792087793825889282597150717, 10.37413804787813767011511561290, 11.33120706085455040414708176564, 12.22124514527243934443952069284, 14.18486076054878851549714353808
