| L(s) = 1 | + (−0.936 − 0.349i)2-s + (−0.565 − 0.308i)3-s + (0.755 + 0.654i)4-s + (−1.72 + 1.41i)5-s + (0.421 + 0.486i)6-s + (1.82 − 2.43i)7-s + (−0.479 − 0.877i)8-s + (−1.39 − 2.17i)9-s + (2.11 − 0.724i)10-s + (1.38 + 0.633i)11-s + (−0.225 − 0.603i)12-s + (3.84 − 2.87i)13-s + (−2.55 + 1.64i)14-s + (1.41 − 0.267i)15-s + (0.142 + 0.989i)16-s + (0.428 − 5.98i)17-s + ⋯ |
| L(s) = 1 | + (−0.662 − 0.247i)2-s + (−0.326 − 0.178i)3-s + (0.377 + 0.327i)4-s + (−0.773 + 0.634i)5-s + (0.172 + 0.198i)6-s + (0.688 − 0.919i)7-s + (−0.169 − 0.310i)8-s + (−0.465 − 0.724i)9-s + (0.669 − 0.228i)10-s + (0.418 + 0.190i)11-s + (−0.0649 − 0.174i)12-s + (1.06 − 0.797i)13-s + (−0.683 + 0.439i)14-s + (0.365 − 0.0691i)15-s + (0.0355 + 0.247i)16-s + (0.103 − 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.537706 - 0.463142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.537706 - 0.463142i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.936 + 0.349i)T \) |
| 5 | \( 1 + (1.72 - 1.41i)T \) |
| 23 | \( 1 + (2.30 + 4.20i)T \) |
| good | 3 | \( 1 + (0.565 + 0.308i)T + (1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (-1.82 + 2.43i)T + (-1.97 - 6.71i)T^{2} \) |
| 11 | \( 1 + (-1.38 - 0.633i)T + (7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-3.84 + 2.87i)T + (3.66 - 12.4i)T^{2} \) |
| 17 | \( 1 + (-0.428 + 5.98i)T + (-16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (0.682 - 0.787i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.893 - 0.774i)T + (4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.384 - 0.112i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.28 + 0.278i)T + (33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (-8.30 - 5.33i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.57 + 6.55i)T + (-23.2 - 36.1i)T^{2} \) |
| 47 | \( 1 + (8.35 - 8.35i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.94 - 5.94i)T + (14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (7.59 + 1.09i)T + (56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.961 + 3.27i)T + (-51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (4.22 - 11.3i)T + (-50.6 - 43.8i)T^{2} \) |
| 71 | \( 1 + (0.865 + 1.89i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.260 + 0.0186i)T + (72.2 - 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.108 + 0.752i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 5.61i)T + (-75.4 - 34.4i)T^{2} \) |
| 89 | \( 1 + (-3.11 + 0.915i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (3.03 + 13.9i)T + (-88.2 + 40.2i)T^{2} \) |
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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
−11.65049237040417169542249016043, −11.08972409992876404514898576867, −10.31079385208405466545798451514, −8.999985833905335927845741520042, −7.947108639016829263543896344990, −7.16470866884781163344309544364, −6.11730462832583218415722647515, −4.28374998455883471561126016608, −3.08678823355008885971295750446, −0.811353789733215503925032086808,
1.75768743870913505068661801876, 3.97292710111567321328304227767, 5.31352515808892332918036083162, 6.22971689507768015483205122539, 7.84598035341654739519252936126, 8.481267992870419048044223377863, 9.170009134360474231307495326191, 10.69805173645300527276988316788, 11.42940049992231475975006213967, 12.00388690711220756055885261779
