| 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
−10.17034915946310110521157569512, −9.444348095909191391936372435168, −8.877342503050836551893283091576, −7.73606284169201834176540145875, −7.64728771664091179709676384226, −6.40165944708687311561074231532, −4.89066326346746235467691758304, −3.72026700628195461633751336516, −2.44665288921417164323066028015, −1.07768485989707330481403823460,
1.20630475651460491342631643291, 2.46362383259818791241538256528, 3.82182955352820187313521274290, 4.78882990202431282068555076961, 6.49619784756345029809315167207, 7.28863057321648908072417067589, 8.247745193878559490368021741998, 8.912020495416270848016624575897, 9.382508290868523915137869396192, 10.20816401322094067716620324271
