| L(s) = 1 | − 13·2-s + 18·3-s + 71·4-s + 58·5-s − 234·6-s − 146·7-s − 65·8-s + 243·9-s − 754·10-s + 1.27e3·12-s + 130·13-s + 1.89e3·14-s + 1.04e3·15-s − 1.72e3·16-s + 728·17-s − 3.15e3·18-s + 828·19-s + 4.11e3·20-s − 2.62e3·21-s − 238·23-s − 1.17e3·24-s − 2.90e3·25-s − 1.69e3·26-s + 2.91e3·27-s − 1.03e4·28-s − 696·29-s − 1.35e4·30-s + ⋯ |
| L(s) = 1 | − 2.29·2-s + 1.15·3-s + 2.21·4-s + 1.03·5-s − 2.65·6-s − 1.12·7-s − 0.359·8-s + 9-s − 2.38·10-s + 2.56·12-s + 0.213·13-s + 2.58·14-s + 1.19·15-s − 1.68·16-s + 0.610·17-s − 2.29·18-s + 0.526·19-s + 2.30·20-s − 1.30·21-s − 0.0938·23-s − 0.414·24-s − 0.928·25-s − 0.490·26-s + 0.769·27-s − 2.49·28-s − 0.153·29-s − 2.75·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 13 T + 49 p T^{2} + 13 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 58 T + 6266 T^{2} - 58 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 146 T + 7230 T^{2} + 146 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 p T + 701634 T^{2} - 10 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 728 T + 1830542 T^{2} - 728 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 828 T - 865114 T^{2} - 828 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 238 T + 11988422 T^{2} + 238 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 24 p T + 22778374 T^{2} + 24 p^{6} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10480 T + 63595902 T^{2} + 10480 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1908 T + 74176718 T^{2} + 1908 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 36484 T + 564482438 T^{2} + 36484 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9768 T + 237972854 T^{2} + 9768 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 43742 T + 935775830 T^{2} - 43742 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12174 T + 457325722 T^{2} + 12174 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2788 T + 1399448534 T^{2} + 2788 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 25302 T + 1815376826 T^{2} - 25302 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 40520 T + 1779236982 T^{2} + 40520 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 31386 T + 3698331094 T^{2} - 31386 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 46780 T + 4437463638 T^{2} - 46780 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16850 T + 5148027246 T^{2} - 16850 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 79440 T + 5477266486 T^{2} + 79440 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 54204 T + 6019532470 T^{2} + 54204 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 241568 T + 30950947518 T^{2} + 241568 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.931655206771552084806713091012, −9.883820223492675200607433298939, −9.328754514455020055031981321521, −9.174324454845463526324381712698, −8.666779750641210600726053071582, −8.280620805488899382633943259512, −7.74054846021476708734896733148, −7.35595740310625924067744004722, −6.84202043545838361420543290960, −6.41145241124622441968015202959, −5.38636802357761254157050554863, −5.36057292057068168494447936873, −3.82510910126143718009682861300, −3.82398335060610076765580127974, −2.84215872493269310599078104036, −2.24712214365111957159368605406, −1.44493855641174245224782831239, −1.36320707161484610447520483394, 0, 0,
1.36320707161484610447520483394, 1.44493855641174245224782831239, 2.24712214365111957159368605406, 2.84215872493269310599078104036, 3.82398335060610076765580127974, 3.82510910126143718009682861300, 5.36057292057068168494447936873, 5.38636802357761254157050554863, 6.41145241124622441968015202959, 6.84202043545838361420543290960, 7.35595740310625924067744004722, 7.74054846021476708734896733148, 8.280620805488899382633943259512, 8.666779750641210600726053071582, 9.174324454845463526324381712698, 9.328754514455020055031981321521, 9.883820223492675200607433298939, 9.931655206771552084806713091012
