| L(s) = 1 | + (1.99 − 0.0804i)2-s + (−0.751 − 2.59i)3-s + (1.98 − 0.160i)4-s + (1.41 − 0.534i)5-s + (−1.70 − 5.11i)6-s + (−1.63 + 3.84i)7-s + (−0.0203 + 0.00246i)8-s + (−3.62 + 2.29i)9-s + (2.77 − 1.18i)10-s + (3.00 − 4.75i)11-s + (−1.90 − 5.02i)12-s + (−0.967 + 3.47i)13-s + (−2.95 + 7.80i)14-s + (−2.44 − 3.25i)15-s + (−3.96 + 0.644i)16-s + (6.45 + 2.74i)17-s + ⋯ |
| L(s) = 1 | + (1.41 − 0.0568i)2-s + (−0.433 − 1.49i)3-s + (0.991 − 0.0800i)4-s + (0.630 − 0.239i)5-s + (−0.697 − 2.08i)6-s + (−0.619 + 1.45i)7-s + (−0.00719 + 0.000873i)8-s + (−1.20 + 0.765i)9-s + (0.876 − 0.373i)10-s + (0.907 − 1.43i)11-s + (−0.550 − 1.45i)12-s + (−0.268 + 0.963i)13-s + (−0.791 + 2.08i)14-s + (−0.631 − 0.840i)15-s + (−0.992 + 0.161i)16-s + (1.56 + 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.71236 - 1.02901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71236 - 1.02901i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (0.967 - 3.47i)T \) |
| good | 2 | \( 1 + (-1.99 + 0.0804i)T + (1.99 - 0.160i)T^{2} \) |
| 3 | \( 1 + (0.751 + 2.59i)T + (-2.53 + 1.60i)T^{2} \) |
| 5 | \( 1 + (-1.41 + 0.534i)T + (3.74 - 3.31i)T^{2} \) |
| 7 | \( 1 + (1.63 - 3.84i)T + (-4.84 - 5.04i)T^{2} \) |
| 11 | \( 1 + (-3.00 + 4.75i)T + (-4.71 - 9.93i)T^{2} \) |
| 17 | \( 1 + (-6.45 - 2.74i)T + (11.7 + 12.2i)T^{2} \) |
| 19 | \( 1 + (1.84 + 1.06i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.789 - 1.36i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.179 + 4.46i)T + (-28.9 + 2.33i)T^{2} \) |
| 31 | \( 1 + (2.69 - 3.04i)T + (-3.73 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-3.77 - 0.771i)T + (34.0 + 14.5i)T^{2} \) |
| 41 | \( 1 + (4.49 - 1.30i)T + (34.6 - 21.9i)T^{2} \) |
| 43 | \( 1 + (0.625 + 3.06i)T + (-39.5 + 16.8i)T^{2} \) |
| 47 | \( 1 + (6.46 + 4.46i)T + (16.6 + 43.9i)T^{2} \) |
| 53 | \( 1 + (0.932 + 7.68i)T + (-51.4 + 12.6i)T^{2} \) |
| 59 | \( 1 + (0.475 - 2.92i)T + (-55.9 - 18.6i)T^{2} \) |
| 61 | \( 1 + (-2.12 - 1.59i)T + (16.9 + 58.5i)T^{2} \) |
| 67 | \( 1 + (-0.307 + 3.81i)T + (-66.1 - 10.7i)T^{2} \) |
| 71 | \( 1 + (0.130 + 0.125i)T + (2.85 + 70.9i)T^{2} \) |
| 73 | \( 1 + (-3.18 - 6.06i)T + (-41.4 + 60.0i)T^{2} \) |
| 79 | \( 1 + (-5.93 + 8.60i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (1.37 + 5.57i)T + (-73.4 + 38.5i)T^{2} \) |
| 89 | \( 1 + (6.07 - 3.50i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.22 - 5.89i)T + (19.4 - 95.0i)T^{2} \) |
| 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
−12.72652660454853744500189203057, −11.89312962740772014836419901823, −11.55772349611556794897175467011, −9.429731256917724867044323322804, −8.426965496619208430276695974318, −6.66802836260685388606747661991, −6.00272546239387744855730302087, −5.47943548666149444024452923773, −3.37399602224962861708691158748, −1.92634983281216017801913512478,
3.28483792683942045924106736024, 4.17120325450993884921021969867, 5.04494784967222355613684942589, 6.14929801836633023127437344918, 7.28684957807178696359444977514, 9.679441812897431866515373034059, 9.890244414111634039605314220123, 10.91174770548741446243362060586, 12.18049988095950996930461552333, 12.99209356429649658438141146108
