| L(s) = 1 | + (−0.421 + 0.571i)2-s + (−0.246 + 0.0466i)3-s + (0.440 + 1.42i)4-s + (2.22 + 0.262i)5-s + (0.0773 − 0.160i)6-s + (2.64 − 0.120i)7-s + (−2.34 − 0.820i)8-s + (−2.73 + 1.07i)9-s + (−1.08 + 1.15i)10-s + (0.529 − 1.34i)11-s + (−0.175 − 0.331i)12-s + (3.28 + 0.370i)13-s + (−1.04 + 1.56i)14-s + (−0.559 + 0.0388i)15-s + (−1.01 + 0.690i)16-s + (−0.630 − 0.0235i)17-s + ⋯ |
| L(s) = 1 | + (−0.298 + 0.404i)2-s + (−0.142 + 0.0269i)3-s + (0.220 + 0.714i)4-s + (0.993 + 0.117i)5-s + (0.0315 − 0.0655i)6-s + (0.998 − 0.0456i)7-s + (−0.828 − 0.289i)8-s + (−0.911 + 0.357i)9-s + (−0.343 + 0.366i)10-s + (0.159 − 0.406i)11-s + (−0.0505 − 0.0957i)12-s + (0.910 + 0.102i)13-s + (−0.279 + 0.417i)14-s + (−0.144 + 0.0100i)15-s + (−0.253 + 0.172i)16-s + (−0.152 − 0.00572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02307 + 0.752733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02307 + 0.752733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.22 - 0.262i)T \) |
| 7 | \( 1 + (-2.64 + 0.120i)T \) |
| good | 2 | \( 1 + (0.421 - 0.571i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (0.246 - 0.0466i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-0.529 + 1.34i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-3.28 - 0.370i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (0.630 + 0.0235i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (1.79 - 3.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0779 - 2.08i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (0.472 - 0.107i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (5.64 - 3.25i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.99 + 3.69i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (4.67 + 9.70i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (4.24 + 12.1i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-1.53 - 1.13i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-1.05 - 0.557i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.818 + 10.9i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (3.36 - 10.9i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-4.90 - 1.31i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.90 + 8.35i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (0.574 - 0.424i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (8.89 + 5.13i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.25 + 11.1i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-3.16 - 8.07i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-6.86 - 6.86i)T + 97iT^{2} \) |
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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
−12.17327565174953096292211043285, −11.24049950565783350153514839445, −10.55493946890192005220020367474, −8.965779335543883632647788019908, −8.527546133928515235295103672279, −7.40628348131077522826873984565, −6.18638171874793030052514195854, −5.38492810982080751450320079634, −3.62975001052423597817956947000, −2.07038022591944037916365140796,
1.35545694658076012940219470133, 2.63733332337251842759318066888, 4.77502228234619983121960694797, 5.81961700527345536521701207216, 6.55591996534890668364474488864, 8.319796317392223304207199935505, 9.127008183001210530996366308169, 10.00584761192644607216704929770, 11.17782204643352514113980634081, 11.37682255114319400341316314292
