| L(s) = 1 | + (0.948 − 1.28i)2-s + (0.585 − 0.110i)3-s + (−0.163 − 0.529i)4-s + (0.453 + 2.18i)5-s + (0.413 − 0.857i)6-s + (2.57 + 0.606i)7-s + (2.18 + 0.763i)8-s + (−2.46 + 0.966i)9-s + (3.24 + 1.49i)10-s + (1.01 − 2.59i)11-s + (−0.154 − 0.291i)12-s + (−6.69 − 0.754i)13-s + (3.22 − 2.73i)14-s + (0.508 + 1.23i)15-s + (3.96 − 2.70i)16-s + (−2.66 − 0.0996i)17-s + ⋯ |
| L(s) = 1 | + (0.670 − 0.909i)2-s + (0.337 − 0.0639i)3-s + (−0.0815 − 0.264i)4-s + (0.202 + 0.979i)5-s + (0.168 − 0.350i)6-s + (0.973 + 0.229i)7-s + (0.771 + 0.269i)8-s + (−0.820 + 0.322i)9-s + (1.02 + 0.472i)10-s + (0.306 − 0.781i)11-s + (−0.0444 − 0.0841i)12-s + (−1.85 − 0.209i)13-s + (0.861 − 0.731i)14-s + (0.131 + 0.317i)15-s + (0.991 − 0.676i)16-s + (−0.646 − 0.0241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96366 - 0.537297i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96366 - 0.537297i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.453 - 2.18i)T \) |
| 7 | \( 1 + (-2.57 - 0.606i)T \) |
| good | 2 | \( 1 + (-0.948 + 1.28i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.585 + 0.110i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-1.01 + 2.59i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (6.69 + 0.754i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (2.66 + 0.0996i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-2.69 + 4.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.209 + 5.61i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (1.09 - 0.250i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (3.55 - 2.05i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.77 + 3.58i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-2.43 - 5.05i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.305 + 0.871i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-3.03 - 2.24i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (2.15 + 1.13i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.773 - 10.3i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-1.50 + 4.87i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (5.02 + 1.34i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.07 - 4.69i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (4.39 - 3.24i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (6.60 + 3.81i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.753 - 6.68i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-5.20 - 13.2i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (10.8 + 10.8i)T + 97iT^{2} \) |
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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
−11.77830034481631109221484670739, −11.24086169953973032075566207397, −10.56686303513864230417415218834, −9.230952429370767735878992878599, −8.009442154286540903369756954968, −7.15116201028064249035950988918, −5.53677657079694452036457570263, −4.48332006800398976921532559855, −2.87021861058741550613296842957, −2.36599013168211068736690957588,
1.87112770264448729269149282470, 4.15182954943523847139187236887, 5.00568862393542990715081115437, 5.80224990811797709105467355663, 7.31564346810850675815299191305, 7.918975168032262626519289439035, 9.218471399331675015373082536697, 9.992438156069233975936708278932, 11.53062249855566396426341256834, 12.29637899277760941414676504065
