| L(s) = 1 | + (−1.69 − 0.148i)2-s + (1.72 + 0.193i)3-s + (0.879 + 0.155i)4-s + (−2.88 − 0.582i)6-s + (−0.290 − 0.203i)7-s + (1.81 + 0.487i)8-s + (2.92 + 0.664i)9-s + (1.06 − 2.91i)11-s + (1.48 + 0.436i)12-s + (0.491 + 5.61i)13-s + (0.461 + 0.387i)14-s + (−4.68 − 1.70i)16-s + (−1.42 + 0.382i)17-s + (−4.85 − 1.56i)18-s + (3.38 − 1.95i)19-s + ⋯ |
| L(s) = 1 | + (−1.19 − 0.104i)2-s + (0.993 + 0.111i)3-s + (0.439 + 0.0775i)4-s + (−1.17 − 0.237i)6-s + (−0.109 − 0.0768i)7-s + (0.642 + 0.172i)8-s + (0.975 + 0.221i)9-s + (0.320 − 0.879i)11-s + (0.428 + 0.126i)12-s + (0.136 + 1.55i)13-s + (0.123 + 0.103i)14-s + (−1.17 − 0.426i)16-s + (−0.346 + 0.0928i)17-s + (−1.14 − 0.367i)18-s + (0.775 − 0.447i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15650 + 0.0670757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15650 + 0.0670757i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 - 0.193i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (1.69 + 0.148i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (0.290 + 0.203i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.06 + 2.91i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.491 - 5.61i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (1.42 - 0.382i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.38 + 1.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0283 - 0.0404i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-5.09 + 4.27i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.975 - 5.53i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.772 - 2.88i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-4.90 + 5.84i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.74 - 8.03i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.549 + 0.785i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-6.69 + 6.69i)T - 53iT^{2} \) |
| 59 | \( 1 + (10.5 - 3.84i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.25 - 7.14i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.00 - 0.175i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 6.62i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.49 + 5.58i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.11 - 8.48i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.433 + 4.95i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-1.68 - 2.91i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.1 + 6.12i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.20816401322094067716620324271, −9.382508290868523915137869396192, −8.912020495416270848016624575897, −8.247745193878559490368021741998, −7.28863057321648908072417067589, −6.49619784756345029809315167207, −4.78882990202431282068555076961, −3.82182955352820187313521274290, −2.46362383259818791241538256528, −1.20630475651460491342631643291,
1.07768485989707330481403823460, 2.44665288921417164323066028015, 3.72026700628195461633751336516, 4.89066326346746235467691758304, 6.40165944708687311561074231532, 7.64728771664091179709676384226, 7.73606284169201834176540145875, 8.877342503050836551893283091576, 9.444348095909191391936372435168, 10.17034915946310110521157569512
