| L(s) = 1 | + (−0.623 − 0.781i)2-s + (−3.32 − 0.500i)3-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (1.67 + 2.90i)6-s + (−0.871 + 1.50i)7-s + (0.900 − 0.433i)8-s + (7.91 + 2.44i)9-s + (−0.0747 − 0.997i)10-s + (−0.296 − 1.30i)11-s + (1.22 − 3.12i)12-s + (0.276 − 3.68i)13-s + (1.72 − 0.259i)14-s + (−2.46 − 2.28i)15-s + (−0.900 − 0.433i)16-s + (0.528 − 0.360i)17-s + ⋯ |
| L(s) = 1 | + (−0.440 − 0.552i)2-s + (−1.91 − 0.289i)3-s + (−0.111 + 0.487i)4-s + (0.369 + 0.251i)5-s + (0.685 + 1.18i)6-s + (−0.329 + 0.570i)7-s + (0.318 − 0.153i)8-s + (2.63 + 0.813i)9-s + (−0.0236 − 0.315i)10-s + (−0.0894 − 0.392i)11-s + (0.354 − 0.902i)12-s + (0.0766 − 1.02i)13-s + (0.460 − 0.0694i)14-s + (−0.635 − 0.589i)15-s + (−0.225 − 0.108i)16-s + (0.128 − 0.0874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0778593 - 0.290425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0778593 - 0.290425i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.623 + 0.781i)T \) |
| 5 | \( 1 + (-0.826 - 0.563i)T \) |
| 43 | \( 1 + (-5.14 + 4.06i)T \) |
| good | 3 | \( 1 + (3.32 + 0.500i)T + (2.86 + 0.884i)T^{2} \) |
| 7 | \( 1 + (0.871 - 1.50i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.296 + 1.30i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.276 + 3.68i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-0.528 + 0.360i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (0.715 - 0.220i)T + (15.6 - 10.7i)T^{2} \) |
| 23 | \( 1 + (5.37 - 4.98i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (2.86 - 0.431i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (-1.79 + 4.56i)T + (-22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (4.24 + 7.34i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.87 + 8.61i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-2.39 + 10.4i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (0.218 + 2.92i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (2.94 + 1.42i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (3.59 + 9.17i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-1.15 + 0.357i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (5.28 + 4.90i)T + (5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (0.768 - 10.2i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (1.81 - 3.14i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.48 - 0.827i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (2.95 + 0.445i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (-2.40 - 10.5i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−10.73445535771768982724240170358, −10.28057315953301314419805081669, −9.331349709463105778865318310620, −7.889221609400652280520236982937, −6.93586512960156137438505199744, −5.78256708626648078414257307945, −5.43469637024197478931103403985, −3.78444158647059366430503361883, −1.97441284008355544235617386760, −0.30175918976881496951447166959,
1.37474770878395691245291736354, 4.25888744583540956695473760537, 4.91365663627872749203901724183, 6.15846896448923283442079182132, 6.53220634287266669781090873279, 7.53722199718811222940431311557, 9.035319882866191079659052187048, 10.08660952768083767024153012622, 10.36280629220408532189976107752, 11.45335820845030335793523022241
