| L(s) = 1 | + (0.135 − 0.398i)2-s + (0.651 + 0.503i)4-s + (0.928 + 0.370i)7-s + (0.639 − 0.423i)8-s + (0.639 − 0.768i)9-s + (−0.151 + 0.979i)11-s + (0.273 − 0.319i)14-s + (0.126 + 0.484i)16-s + (−0.219 − 0.359i)18-s + (0.369 + 0.193i)22-s + (−0.305 − 1.24i)23-s + (−0.628 + 0.777i)25-s + (0.418 + 0.708i)28-s + (0.276 − 0.00808i)29-s + (0.974 + 0.0713i)32-s + ⋯ |
| L(s) = 1 | + (0.135 − 0.398i)2-s + (0.651 + 0.503i)4-s + (0.928 + 0.370i)7-s + (0.639 − 0.423i)8-s + (0.639 − 0.768i)9-s + (−0.151 + 0.979i)11-s + (0.273 − 0.319i)14-s + (0.126 + 0.484i)16-s + (−0.219 − 0.359i)18-s + (0.369 + 0.193i)22-s + (−0.305 − 1.24i)23-s + (−0.628 + 0.777i)25-s + (0.418 + 0.708i)28-s + (0.276 − 0.00808i)29-s + (0.974 + 0.0713i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.872371973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.872371973\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.928 - 0.370i)T \) |
| 431 | \( 1 + (-0.989 - 0.145i)T \) |
| good | 2 | \( 1 + (-0.135 + 0.398i)T + (-0.791 - 0.611i)T^{2} \) |
| 3 | \( 1 + (-0.639 + 0.768i)T^{2} \) |
| 5 | \( 1 + (0.628 - 0.777i)T^{2} \) |
| 11 | \( 1 + (0.151 - 0.979i)T + (-0.953 - 0.302i)T^{2} \) |
| 13 | \( 1 + (-0.336 - 0.941i)T^{2} \) |
| 17 | \( 1 + (0.00730 + 0.999i)T^{2} \) |
| 19 | \( 1 + (-0.965 - 0.259i)T^{2} \) |
| 23 | \( 1 + (0.305 + 1.24i)T + (-0.885 + 0.463i)T^{2} \) |
| 29 | \( 1 + (-0.276 + 0.00808i)T + (0.998 - 0.0584i)T^{2} \) |
| 31 | \( 1 + (-0.661 - 0.749i)T^{2} \) |
| 37 | \( 1 + (0.868 + 0.361i)T + (0.704 + 0.709i)T^{2} \) |
| 41 | \( 1 + (-0.993 - 0.116i)T^{2} \) |
| 43 | \( 1 + (1.15 + 0.135i)T + (0.972 + 0.231i)T^{2} \) |
| 47 | \( 1 + (-0.833 - 0.551i)T^{2} \) |
| 53 | \( 1 + (0.881 + 1.27i)T + (-0.350 + 0.936i)T^{2} \) |
| 59 | \( 1 + (-0.281 - 0.959i)T^{2} \) |
| 61 | \( 1 + (-0.849 - 0.527i)T^{2} \) |
| 67 | \( 1 + (0.158 - 1.96i)T + (-0.987 - 0.160i)T^{2} \) |
| 71 | \( 1 + (1.29 - 0.450i)T + (0.782 - 0.622i)T^{2} \) |
| 73 | \( 1 + (0.557 + 0.829i)T^{2} \) |
| 79 | \( 1 + (-0.175 + 0.762i)T + (-0.899 - 0.437i)T^{2} \) |
| 83 | \( 1 + (-0.928 - 0.370i)T^{2} \) |
| 89 | \( 1 + (0.672 - 0.739i)T^{2} \) |
| 97 | \( 1 + (-0.470 + 0.882i)T^{2} \) |
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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
−8.795263159024143209544041221310, −8.152114144132206154139112493016, −7.28487988903309203199831719778, −6.87112414480073125484807274445, −5.90200144178374051808131601722, −4.80513731703591196333291753994, −4.19048967276632624361780355730, −3.27203212993400300570999535363, −2.18052105883770937977790668109, −1.53006824165032809978861341433,
1.34701528615595090729980213673, 2.08342369637316465622700653951, 3.33182111363218332542523855793, 4.48969520185599778948166862006, 5.11661036469012109180365445127, 5.86132601766941759864110874655, 6.62745520290725657607072200148, 7.59653328910258093419248886584, 7.83987389856391934189814362584, 8.685408548283583268487255443298
