| L(s) = 1 | + (0.146 − 1.95i)2-s + (1.83 − 1.25i)3-s + (−1.82 − 0.275i)4-s + (−0.397 + 1.74i)5-s + (−2.18 − 3.77i)6-s + (1.77 − 1.96i)7-s + (0.0666 − 0.291i)8-s + (0.709 − 1.80i)9-s + (3.35 + 1.03i)10-s + (−0.800 + 2.04i)11-s + (−3.70 + 1.78i)12-s + (−1.67 − 1.55i)13-s + (−3.57 − 3.75i)14-s + (1.45 + 3.69i)15-s + (−4.09 − 1.26i)16-s + (−0.0840 − 0.368i)17-s + ⋯ |
| L(s) = 1 | + (0.103 − 1.38i)2-s + (1.06 − 0.723i)3-s + (−0.913 − 0.137i)4-s + (−0.177 + 0.779i)5-s + (−0.890 − 1.54i)6-s + (0.670 − 0.741i)7-s + (0.0235 − 0.103i)8-s + (0.236 − 0.602i)9-s + (1.05 + 0.326i)10-s + (−0.241 + 0.615i)11-s + (−1.06 + 0.514i)12-s + (−0.465 − 0.432i)13-s + (−0.956 − 1.00i)14-s + (0.374 + 0.955i)15-s + (−1.02 − 0.315i)16-s + (−0.0203 − 0.0892i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.775029 - 1.71650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.775029 - 1.71650i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.77 + 1.96i)T \) |
| 43 | \( 1 + (3.71 - 5.40i)T \) |
| good | 2 | \( 1 + (-0.146 + 1.95i)T + (-1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (-1.83 + 1.25i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.397 - 1.74i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (0.800 - 2.04i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (1.67 + 1.55i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.0840 + 0.368i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (0.0562 - 0.0705i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (-1.42 - 1.79i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (0.412 - 5.50i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (0.133 - 1.78i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 - 0.117T + 37T^{2} \) |
| 41 | \( 1 + (-3.29 + 1.58i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (1.79 + 4.58i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-9.45 - 2.91i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (5.77 - 5.36i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (0.873 + 11.6i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (1.32 + 3.37i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (-2.64 - 6.74i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (0.0227 - 0.0994i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (7.96 + 13.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.931 - 12.4i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (11.0 + 5.33i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (8.65 + 10.8i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.28417626345650893117762161226, −10.63893932726950438055219649153, −9.802753014733478063004637249789, −8.622729983163307129009151272601, −7.44521795741012369078865540091, −7.03914004047646013732163508928, −4.84549769637585798733399672118, −3.50228641777884031878679285248, −2.64874425876529481391357129796, −1.51998917525929835711945847282,
2.50140918132230340556707629315, 4.23595586050420658226203646588, 5.04754714618752031729188896003, 6.08594160746343010377912591205, 7.50466730117791443038668377811, 8.510163300073893602765469956879, 8.665814926400872454529526670539, 9.704034491559559419454748492864, 11.14421052265830961132988507700, 12.17915082945500769716627373823
