L(s) = 1 | + (0.134 + 3.99i)2-s + (9.27 − 9.27i)3-s + (−15.9 + 1.07i)4-s + (21.1 − 13.3i)5-s + (38.3 + 35.8i)6-s + (26.2 − 26.2i)7-s + (−6.46 − 63.6i)8-s − 90.9i·9-s + (56.2 + 82.6i)10-s + 212. i·11-s + (−138. + 158. i)12-s + (35.2 − 35.2i)13-s + (108. + 101. i)14-s + (72.1 − 319. i)15-s + (253. − 34.4i)16-s + (−128. + 128. i)17-s + ⋯ |
L(s) = 1 | + (0.0337 + 0.999i)2-s + (1.03 − 1.03i)3-s + (−0.997 + 0.0674i)4-s + (0.845 − 0.534i)5-s + (1.06 + 0.995i)6-s + (0.536 − 0.536i)7-s + (−0.101 − 0.994i)8-s − 1.12i·9-s + (0.562 + 0.826i)10-s + 1.75i·11-s + (−0.958 + 1.09i)12-s + (0.208 − 0.208i)13-s + (0.553 + 0.517i)14-s + (0.320 − 1.42i)15-s + (0.990 − 0.134i)16-s + (−0.444 + 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.05042 + 0.114210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.05042 + 0.114210i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.134 - 3.99i)T \) |
| 5 | \( 1 + (-21.1 + 13.3i)T \) |
good | 3 | \( 1 + (-9.27 + 9.27i)T - 81iT^{2} \) |
| 7 | \( 1 + (-26.2 + 26.2i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 - 212. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-35.2 + 35.2i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (128. - 128. i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 283.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (219. + 219. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 - 304.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.69e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-1.55e3 - 1.55e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 247.T + 2.82e6T^{2} \) |
| 43 | \( 1 + (2.28e3 - 2.28e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-510. + 510. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.01e3 + 1.01e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 3.14e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 3.79e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + (413. + 413. i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 937.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (4.21e3 + 4.21e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 719. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-726. + 726. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.16e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.56e3 + 1.56e3i)T - 8.85e7iT^{2} \) |
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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
−15.00796099701452730592073961606, −14.34000306692426687713760348542, −13.17599493543944254273027275767, −12.70060086187679959635418646067, −9.991185172025438248287834553568, −8.720408026574279157530531280435, −7.70026757408127275859911659167, −6.54841016058065918357259120413, −4.60432872539886069948841696515, −1.74763420893932003897608441522,
2.38157833328713891074636289669, 3.71202478829666236667575899170, 5.56160629507116507654517216870, 8.538687624037375172322024990792, 9.204051078658884998275501394718, 10.48557375442547635265176976463, 11.36682039731463242378898941644, 13.34103845078999396796384680809, 14.15171586113279085630841963450, 14.93692118222737884329113880587