| L(s) = 1 | + (−2.52 − 0.220i)2-s + (3.65 − 3.69i)3-s + (−1.55 − 0.274i)4-s + (−0.664 − 11.1i)5-s + (−10.0 + 8.52i)6-s + (29.5 + 20.7i)7-s + (23.4 + 6.28i)8-s + (−0.350 − 26.9i)9-s + (−0.787 + 28.3i)10-s + (4.71 − 12.9i)11-s + (−6.69 + 4.75i)12-s + (−3.28 − 37.6i)13-s + (−70.0 − 58.7i)14-s + (−43.6 − 38.2i)15-s + (−45.9 − 16.7i)16-s + (42.8 − 11.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.892 − 0.0780i)2-s + (0.702 − 0.711i)3-s + (−0.194 − 0.0343i)4-s + (−0.0594 − 0.998i)5-s + (−0.682 + 0.580i)6-s + (1.59 + 1.11i)7-s + (1.03 + 0.277i)8-s + (−0.0129 − 0.999i)9-s + (−0.0248 + 0.895i)10-s + (0.129 − 0.354i)11-s + (−0.161 + 0.114i)12-s + (−0.0701 − 0.802i)13-s + (−1.33 − 1.12i)14-s + (−0.752 − 0.658i)15-s + (−0.717 − 0.261i)16-s + (0.611 − 0.163i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.845436 - 0.987441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.845436 - 0.987441i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.65 + 3.69i)T \) |
| 5 | \( 1 + (0.664 + 11.1i)T \) |
| good | 2 | \( 1 + (2.52 + 0.220i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-29.5 - 20.7i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (-4.71 + 12.9i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (3.28 + 37.6i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (-42.8 + 11.4i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (59.3 - 34.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (68.5 + 97.9i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (-142. + 119. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-16.8 + 95.3i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-53.4 - 199. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-34.3 + 40.9i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (72.8 + 156. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (180. - 257. i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (-422. + 422. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-122. + 44.5i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-127. - 724. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (294. - 25.7i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (588. + 339. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-149. + 559. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (-209. - 249. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (131. - 1.49e3i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-97.6 - 169. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.03e3 + 482. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.38354542551818066231436973650, −11.62559521492249141639671408498, −10.10265981944832280568938915909, −8.890504075116355845500477598075, −8.266868869627429192514456468140, −7.917691967460806168244119161308, −5.75244515552492502996266836528, −4.50753677675416663619654013536, −2.13570675414226854694372824023, −0.916366768074382183349176889788,
1.75711294329166800627413218947, 3.86822051609226308244371049941, 4.75751239096156674756053020823, 7.16097617642466123725245773862, 7.84087968895541804456501724397, 8.797333558501421316014357790527, 10.04660116560105321291792774028, 10.59247245275491650652016233684, 11.49104092345421063636194968022, 13.50770898958290602026363419408
