L(s) = 1 | + (−18.5 + 0.866i)7-s + 89·13-s + (81.5 − 141. i)19-s + (62.5 + 108. i)25-s + (9.5 + 16.4i)31-s + (216.5 − 374. i)37-s + 449·43-s + (341.5 − 32.0i)49-s + (−91 + 157. i)61-s + (−503.5 − 872. i)67-s + (459.5 + 795. i)73-s + (−251.5 + 435. i)79-s + (−1.64e3 + 77.0i)91-s − 1.33e3·97-s + (9.5 − 16.4i)103-s + ⋯ |
L(s) = 1 | + (−0.998 + 0.0467i)7-s + 1.89·13-s + (0.984 − 1.70i)19-s + (0.5 + 0.866i)25-s + (0.0550 + 0.0953i)31-s + (0.961 − 1.66i)37-s + 1.59·43-s + (0.995 − 0.0934i)49-s + (−0.191 + 0.330i)61-s + (−0.918 − 1.59i)67-s + (0.736 + 1.27i)73-s + (−0.358 + 0.620i)79-s + (−1.89 + 0.0887i)91-s − 1.39·97-s + (0.00908 − 0.0157i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.702673961\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.702673961\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (18.5 - 0.866i)T \) |
good | 5 | \( 1 + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 89T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-81.5 + 141. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + (-9.5 - 16.4i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-216.5 + 374. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 449T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (91 - 157. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (503.5 + 872. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + (-459.5 - 795. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (251.5 - 435. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−11.34378914664148130294773976324, −10.76707851053638031530109253656, −9.387745442176583721619123490349, −8.901575701749760013159730014988, −7.47715849709593328140660720171, −6.48122715099855660866464534333, −5.51690785068043219668600762510, −3.95433941747557553840356413728, −2.86341616661481952328288138261, −0.874151340891841604116570156248,
1.14117738000301919970041639552, 3.08530979194786457608277993893, 4.07502494686584631679024140890, 5.80619511458025153892959648891, 6.43275887400981311265324539650, 7.80222347106362064644519515669, 8.763820095806108690421341583587, 9.817189925845689694851714872995, 10.61359898254344370671068778424, 11.70590681576842493966553960138