| L(s) = 1 | + (3.61 − 2.62i)2-s + (−2.30 − 7.09i)3-s + (3.68 − 11.3i)4-s + (6.84 + 4.97i)5-s + (−26.9 − 19.5i)6-s + (6.25 − 19.2i)7-s + (−5.41 − 16.6i)8-s + (−23.2 + 16.8i)9-s + 37.7·10-s − 89.0·12-s + (−49.7 + 36.1i)13-s + (−27.9 − 85.9i)14-s + (19.5 − 60.0i)15-s + (13.8 + 10.0i)16-s + (56.1 + 40.7i)17-s + (−39.6 + 121. i)18-s + ⋯ |
| L(s) = 1 | + (1.27 − 0.927i)2-s + (−0.443 − 1.36i)3-s + (0.460 − 1.41i)4-s + (0.612 + 0.444i)5-s + (−1.83 − 1.33i)6-s + (0.337 − 1.03i)7-s + (−0.239 − 0.737i)8-s + (−0.860 + 0.625i)9-s + 1.19·10-s − 2.14·12-s + (−1.06 + 0.771i)13-s + (−0.533 − 1.64i)14-s + (0.336 − 1.03i)15-s + (0.216 + 0.157i)16-s + (0.800 + 0.581i)17-s + (−0.518 + 1.59i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.785744 - 2.85688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.785744 - 2.85688i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (-3.61 + 2.62i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (2.30 + 7.09i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-6.84 - 4.97i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-6.25 + 19.2i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (49.7 - 36.1i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-56.1 - 40.7i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-2.21 - 6.82i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 50.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (44.2 - 136. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-206. + 149. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-104. + 320. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-55.0 - 169. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 55.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + (79.2 + 243. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (172. - 125. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (246. - 758. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (136. + 99.1i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 366.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-632. - 459. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (295. - 910. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (473. - 344. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-531. - 385. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 72.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-792. + 575. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−12.59254060332427121033478451986, −11.84973061393278171193471856207, −10.90630249319893724843112874113, −9.942723237294809094831814231944, −7.74438783073808592527868637787, −6.73863588461508469844936316556, −5.66209741574237676512220191175, −4.24713587040814953905932295045, −2.48201143873442216068004536635, −1.28847023309359958801345244142,
3.09350327831957234140966288127, 4.84627071959134129176094834496, 5.14114294867063498258760467651, 6.11799874272486815909018577100, 7.81925852131764403300582905018, 9.327289934343473117743970048216, 10.14090323070718643975450382857, 11.67958684360140284192420367014, 12.49379209694435501378778578223, 13.62048220148228325633224086495
