| L(s) = 1 | + (−0.0713 − 0.997i)2-s + (0.977 + 0.212i)3-s + (−0.989 + 0.142i)4-s + (−1.00 − 1.99i)5-s + (0.142 − 0.989i)6-s + (−2.61 − 1.43i)7-s + (0.212 + 0.977i)8-s + (0.909 + 0.415i)9-s + (−1.92 + 1.14i)10-s + (1.89 − 1.64i)11-s + (−0.997 − 0.0713i)12-s + (−0.269 − 0.493i)13-s + (−1.23 + 2.71i)14-s + (−0.553 − 2.16i)15-s + (0.959 − 0.281i)16-s + (−5.07 + 3.79i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 − 0.705i)2-s + (0.564 + 0.122i)3-s + (−0.494 + 0.0711i)4-s + (−0.447 − 0.894i)5-s + (0.0580 − 0.404i)6-s + (−0.990 − 0.540i)7-s + (0.0751 + 0.345i)8-s + (0.303 + 0.138i)9-s + (−0.608 + 0.360i)10-s + (0.570 − 0.494i)11-s + (−0.287 − 0.0205i)12-s + (−0.0747 − 0.136i)13-s + (−0.331 + 0.725i)14-s + (−0.142 − 0.559i)15-s + (0.239 − 0.0704i)16-s + (−1.23 + 0.920i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0295703 + 0.819298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0295703 + 0.819298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 + 0.997i)T \) |
| 3 | \( 1 + (-0.977 - 0.212i)T \) |
| 5 | \( 1 + (1.00 + 1.99i)T \) |
| 23 | \( 1 + (-0.201 - 4.79i)T \) |
| good | 7 | \( 1 + (2.61 + 1.43i)T + (3.78 + 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.89 + 1.64i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.269 + 0.493i)T + (-7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (5.07 - 3.79i)T + (4.78 - 16.3i)T^{2} \) |
| 19 | \( 1 + (1.00 + 6.97i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (8.91 + 1.28i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.85 + 2.47i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.03 + 1.50i)T + (27.9 - 24.2i)T^{2} \) |
| 41 | \( 1 + (2.06 + 4.51i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (0.109 - 0.505i)T + (-39.1 - 17.8i)T^{2} \) |
| 47 | \( 1 + (3.07 - 3.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.79 + 12.4i)T + (-28.6 - 44.5i)T^{2} \) |
| 59 | \( 1 + (-1.37 + 4.67i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.62 + 2.52i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-15.5 + 1.11i)T + (66.3 - 9.53i)T^{2} \) |
| 71 | \( 1 + (6.74 - 7.77i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-6.31 + 8.43i)T + (-20.5 - 70.0i)T^{2} \) |
| 79 | \( 1 + (-0.503 - 0.147i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.02 - 2.74i)T + (-62.7 + 54.3i)T^{2} \) |
| 89 | \( 1 + (-8.73 + 5.61i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (1.87 - 5.02i)T + (-73.3 - 63.5i)T^{2} \) |
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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
−9.820400870444052665275440980906, −9.196301327013961182739117380984, −8.650996506164848428483568822181, −7.57994173650927479981831509618, −6.56244662162677067951380787448, −5.21953470081000128656242189518, −4.00015283724187164756711586078, −3.57757095598306669688967728770, −2.04397237333061681240575301055, −0.39369943839260220202647645268,
2.26909466542605359317651229433, 3.44780313302346099611875131249, 4.33915946097443360575672463364, 5.86333493960181395742081618368, 6.70449713402251448590586314712, 7.22879082353559700753975759266, 8.252730159528567305443472335167, 9.172008374164726220140094090929, 9.758631494057306976198991979521, 10.74954909871241270576452687148
