| L(s) = 1 | + (−0.786 − 0.618i)2-s + (−0.974 + 1.36i)3-s + (0.235 + 0.971i)4-s + (−2.94 + 0.566i)5-s + (1.61 − 0.473i)6-s + (−0.384 − 2.61i)7-s + (0.415 − 0.909i)8-s + (0.0574 + 0.166i)9-s + (2.66 + 1.37i)10-s + (0.100 − 0.0793i)11-s + (−1.56 − 0.624i)12-s + (2.79 − 1.79i)13-s + (−1.31 + 2.29i)14-s + (2.09 − 4.57i)15-s + (−0.888 + 0.458i)16-s + (−4.06 − 3.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.437i)2-s + (−0.562 + 0.790i)3-s + (0.117 + 0.485i)4-s + (−1.31 + 0.253i)5-s + (0.658 − 0.193i)6-s + (−0.145 − 0.989i)7-s + (0.146 − 0.321i)8-s + (0.0191 + 0.0553i)9-s + (0.841 + 0.433i)10-s + (0.0304 − 0.0239i)11-s + (−0.450 − 0.180i)12-s + (0.775 − 0.498i)13-s + (−0.351 + 0.613i)14-s + (0.539 − 1.18i)15-s + (−0.222 + 0.114i)16-s + (−0.985 − 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.375859 - 0.310877i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375859 - 0.310877i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (0.384 + 2.61i)T \) |
| 23 | \( 1 + (-4.43 - 1.82i)T \) |
| good | 3 | \( 1 + (0.974 - 1.36i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (2.94 - 0.566i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-0.100 + 0.0793i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.79 + 1.79i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.06 + 3.87i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.83 + 5.56i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-6.69 + 1.96i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.91 - 0.278i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (3.61 + 10.4i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (7.28 + 8.40i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.260 - 0.569i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (5.01 - 8.68i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.138 + 2.90i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-2.74 - 1.41i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-5.89 - 8.27i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-5.76 + 2.30i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.316 - 2.20i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.43 + 10.0i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.0649 - 1.36i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-1.61 + 1.86i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (1.62 + 0.154i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (10.3 + 12.0i)T + (-13.8 + 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.08996075095644475942810671527, −10.84174093128224457282437132044, −9.740306325980944944772377279805, −8.735681179909472626776104685530, −7.51173861770172544703969561211, −6.98553064274769477993808465628, −5.09392999357244976310219432635, −4.11597912549478867495396644263, −3.16514111452787840935202214454, −0.49974557220739725878341938561,
1.39563926099226753468181022842, 3.54774366828994318443198932150, 5.07501802384922485117736573670, 6.31904080716083987170568053330, 6.91086269140266095326469272483, 8.209732960355775141188093833085, 8.582963580031857838181797672652, 9.851492529210629745034170738842, 11.22767511947241086327869592189, 11.79499606049634499672177492253
