| L(s) = 1 | + (−2.41 − 0.0974i)2-s + (−0.437 + 1.51i)3-s + (3.84 + 0.310i)4-s + (−2.71 − 1.03i)5-s + (1.20 − 3.60i)6-s + (−0.958 − 2.25i)7-s + (−4.46 − 0.541i)8-s + (0.446 + 0.282i)9-s + (6.47 + 2.75i)10-s + (0.842 + 1.33i)11-s + (−2.15 + 5.66i)12-s + (2.05 − 2.96i)13-s + (2.09 + 5.53i)14-s + (2.74 − 3.65i)15-s + (3.12 + 0.507i)16-s + (1.77 − 0.757i)17-s + ⋯ |
| L(s) = 1 | + (−1.70 − 0.0689i)2-s + (−0.252 + 0.871i)3-s + (1.92 + 0.155i)4-s + (−1.21 − 0.460i)5-s + (0.491 − 1.47i)6-s + (−0.362 − 0.850i)7-s + (−1.57 − 0.191i)8-s + (0.148 + 0.0941i)9-s + (2.04 + 0.871i)10-s + (0.253 + 0.401i)11-s + (−0.620 + 1.63i)12-s + (0.570 − 0.821i)13-s + (0.561 + 1.47i)14-s + (0.708 − 0.943i)15-s + (0.780 + 0.126i)16-s + (0.431 − 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.349518 - 0.122549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.349518 - 0.122549i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.05 + 2.96i)T \) |
| good | 2 | \( 1 + (2.41 + 0.0974i)T + (1.99 + 0.160i)T^{2} \) |
| 3 | \( 1 + (0.437 - 1.51i)T + (-2.53 - 1.60i)T^{2} \) |
| 5 | \( 1 + (2.71 + 1.03i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (0.958 + 2.25i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (-0.842 - 1.33i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (-1.77 + 0.757i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (-6.36 + 3.67i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.38 + 4.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.291 + 7.23i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-3.48 - 3.93i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (9.28 - 1.89i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (1.54 + 0.447i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (-0.644 + 3.15i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (0.644 - 0.444i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (-1.39 + 11.4i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (0.344 + 2.12i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (7.39 - 5.55i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (0.146 + 1.81i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (-8.24 + 7.91i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (3.61 - 6.88i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (-3.01 - 4.36i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (-0.682 + 2.76i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (-10.7 - 6.18i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.8 + 8.83i)T + (19.4 + 95.0i)T^{2} \) |
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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
−12.14221014066028751488585518539, −11.27622513998618652106785762210, −10.39667222856493729649801979041, −9.794646601151361824396057739277, −8.678407938003848293348835979369, −7.72621608912310840406925122175, −6.94549624060574498920724531652, −4.84924907516371549306694881196, −3.50354183801910220170761603926, −0.71490873714128110135936236676,
1.37423685818158815826927494890, 3.38543480092005756570974743923, 6.02771091501194485747648972905, 7.09011253873719660935358268936, 7.68816481187188807312820712944, 8.746063285415544060199633057213, 9.637842160647554949716617935933, 10.95126186009872705253737149509, 11.79073495293906981508603973367, 12.23940383048358006306537849463
