| 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
−10.74954909871241270576452687148, −9.758631494057306976198991979521, −9.172008374164726220140094090929, −8.252730159528567305443472335167, −7.22879082353559700753975759266, −6.70449713402251448590586314712, −5.86333493960181395742081618368, −4.33915946097443360575672463364, −3.44780313302346099611875131249, −2.26909466542605359317651229433,
0.39369943839260220202647645268, 2.04397237333061681240575301055, 3.57757095598306669688967728770, 4.00015283724187164756711586078, 5.21953470081000128656242189518, 6.56244662162677067951380787448, 7.57994173650927479981831509618, 8.650996506164848428483568822181, 9.196301327013961182739117380984, 9.820400870444052665275440980906
