| 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 | \( 1 + (7.22 + 5.89i)T + (19.4 + 95.0i)T^{2} \) |
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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
−12.99209356429649658438141146108, −12.18049988095950996930461552333, −10.91174770548741446243362060586, −9.890244414111634039605314220123, −9.679441812897431866515373034059, −7.28684957807178696359444977514, −6.14929801836633023127437344918, −5.04494784967222355613684942589, −4.17120325450993884921021969867, −3.28483792683942045924106736024,
1.92634983281216017801913512478, 3.37399602224962861708691158748, 5.47943548666149444024452923773, 6.00272546239387744855730302087, 6.66802836260685388606747661991, 8.426965496619208430276695974318, 9.429731256917724867044323322804, 11.55772349611556794897175467011, 11.89312962740772014836419901823, 12.72652660454853744500189203057
