L(s) = 1 | + (−9.18 − 5.30i)2-s + (−6.26 − 10.8i)3-s + (40.2 + 69.7i)4-s + (63.9 − 36.9i)5-s + 132. i·6-s + (113. + 196. i)7-s − 514. i·8-s + (43.0 − 74.5i)9-s − 783.·10-s + 134.·11-s + (504. − 873. i)12-s + (−145. + 84.0i)13-s − 2.40e3i·14-s + (−801. − 462. i)15-s + (−1.44e3 + 2.49e3i)16-s + (444. + 256. i)17-s + ⋯ |
L(s) = 1 | + (−1.62 − 0.937i)2-s + (−0.401 − 0.695i)3-s + (1.25 + 2.17i)4-s + (1.14 − 0.660i)5-s + 1.50i·6-s + (0.873 + 1.51i)7-s − 2.84i·8-s + (0.177 − 0.306i)9-s − 2.47·10-s + 0.335·11-s + (1.01 − 1.75i)12-s + (−0.238 + 0.137i)13-s − 3.27i·14-s + (−0.919 − 0.530i)15-s + (−1.40 + 2.43i)16-s + (0.372 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0125 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0125 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.619266 - 0.627070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.619266 - 0.627070i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-8.08e3 - 1.99e3i)T \) |
good | 2 | \( 1 + (9.18 + 5.30i)T + (16 + 27.7i)T^{2} \) |
| 3 | \( 1 + (6.26 + 10.8i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (-63.9 + 36.9i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-113. - 196. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 - 134.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (145. - 84.0i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-444. - 256. i)T + (7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.94e3 + 1.12e3i)T + (1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 - 2.76e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 8.02e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.03e3iT - 2.86e7T^{2} \) |
| 41 | \( 1 + (3.29e3 + 5.70e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + 6.09e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.38e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (6.00e3 - 1.04e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.31e4 - 1.33e4i)T + (3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.92e4 - 1.68e4i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.01e3 + 3.49e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-1.01e4 - 1.75e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + 1.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-2.97e4 + 1.71e4i)T + (1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-4.52e4 + 7.83e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (2.72e4 + 1.57e4i)T + (2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 5.36e4iT - 8.58e9T^{2} \) |
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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
−15.45569594743481432784628905928, −13.29666407880505133156552898658, −12.01229831004486443717776887237, −11.60675070030405693496787762236, −9.658672412093381782003315104748, −9.088786123280708836174186749567, −7.68738234186277262205260740657, −5.81675470231880523636328402874, −2.22255688070132265009620831515, −1.15513126102944802383619004536,
1.34761232518231710062361828052, 5.20283219076777099000984221765, 6.76334993653047251087005188343, 7.83322307009337854097323333749, 9.628931282669784605742101165759, 10.37199731883439722950087562904, 10.98081257419867722123218597681, 14.04350686389290501574106998563, 14.61867043030995161453860132187, 16.32234272316479206054866303204