| L(s) = 1 | + (0.0713 − 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (−1.69 + 1.45i)5-s + (−0.142 − 0.989i)6-s + (0.277 − 0.151i)7-s + (−0.212 + 0.977i)8-s + (0.909 − 0.415i)9-s + (1.32 + 1.79i)10-s + (2.15 + 1.87i)11-s + (−0.997 + 0.0713i)12-s + (−3.11 + 5.70i)13-s + (−0.131 − 0.287i)14-s + (−1.35 + 1.78i)15-s + (0.959 + 0.281i)16-s + (5.95 + 4.45i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.759 + 0.650i)5-s + (−0.0580 − 0.404i)6-s + (0.104 − 0.0572i)7-s + (−0.0751 + 0.345i)8-s + (0.303 − 0.138i)9-s + (0.420 + 0.568i)10-s + (0.650 + 0.563i)11-s + (−0.287 + 0.0205i)12-s + (−0.863 + 1.58i)13-s + (−0.0350 − 0.0768i)14-s + (−0.348 + 0.460i)15-s + (0.239 + 0.0704i)16-s + (1.44 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45241 + 0.268936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45241 + 0.268936i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.69 - 1.45i)T \) |
| 23 | \( 1 + (2.90 + 3.81i)T \) |
| good | 7 | \( 1 + (-0.277 + 0.151i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-2.15 - 1.87i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.11 - 5.70i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-5.95 - 4.45i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.0850 - 0.591i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-6.12 + 0.880i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (6.57 - 4.22i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-6.24 - 2.33i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-2.04 + 4.48i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.249 - 1.14i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-7.17 - 7.17i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.12 + 2.05i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-0.454 - 1.54i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.00440 + 0.00686i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.04 - 0.0750i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (7.43 + 8.57i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-3.18 - 4.25i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (11.4 - 3.35i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.15 + 8.46i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (7.74 + 4.97i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.01 - 2.71i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.46516400118248100825670811368, −9.780514430951185887921834255192, −8.900554183893564684010605575066, −7.938762821709412829822094859239, −7.17405151459014169232434491079, −6.20563700513027587498490337115, −4.50094355473440765507094191120, −3.95135525991679591514834138072, −2.79728931215292335943375889930, −1.61969797572811164028535383711,
0.78284804594931059169243025439, 2.99460395952223336565667825496, 3.90850552716544291269952191833, 5.07478882236614142046229465810, 5.73766378140567043314443701807, 7.30322750874172800159949804667, 7.76758767411509176974150382892, 8.488085176616184464378683163498, 9.423535852788110430809379954082, 10.08070297679941295064263219329
