| L(s) = 1 | + 8·2-s − 26·3-s + 48·4-s − 50·5-s − 208·6-s + 160·7-s + 256·8-s + 256·9-s − 400·10-s − 726·11-s − 1.24e3·12-s + 338·13-s + 1.28e3·14-s + 1.30e3·15-s + 1.28e3·16-s − 2.85e3·17-s + 2.04e3·18-s − 938·19-s − 2.40e3·20-s − 4.16e3·21-s − 5.80e3·22-s − 4.42e3·23-s − 6.65e3·24-s + 1.87e3·25-s + 2.70e3·26-s − 2.05e3·27-s + 7.68e3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.66·3-s + 3/2·4-s − 0.894·5-s − 2.35·6-s + 1.23·7-s + 1.41·8-s + 1.05·9-s − 1.26·10-s − 1.80·11-s − 2.50·12-s + 0.554·13-s + 1.74·14-s + 1.49·15-s + 5/4·16-s − 2.39·17-s + 1.48·18-s − 0.596·19-s − 1.34·20-s − 2.05·21-s − 2.55·22-s − 1.74·23-s − 2.35·24-s + 3/5·25-s + 0.784·26-s − 0.542·27-s + 1.85·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 26 T + 140 p T^{2} + 26 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 160 T + 31554 T^{2} - 160 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 p^{2} T + 400996 T^{2} + 6 p^{7} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 168 p T + 4802758 T^{2} + 168 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 938 T - 768876 T^{2} + 938 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4422 T + 10398892 T^{2} + 4422 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4692 T + 45840754 T^{2} + 4692 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6130 T + 61574412 T^{2} - 6130 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9584 T + 90045738 T^{2} + 9584 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 20040 T + 300633142 T^{2} + 20040 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10006 T + 297471780 T^{2} - 10006 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10512 T + 433516690 T^{2} + 10512 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5448 T + 828168622 T^{2} - 5448 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 22398 T + 625577764 T^{2} + 22398 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12988 T + 759132978 T^{2} - 12988 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 56540 T + 3484519674 T^{2} + 56540 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 69810 T + 4151656852 T^{2} + 69810 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 99832 T + 6264664242 T^{2} - 99832 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 31652 T + 6382570614 T^{2} + 31652 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 69120 T + 8971961626 T^{2} - 69120 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 141612 T + 14646321574 T^{2} + 141612 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7676 T - 3083321082 T^{2} + 7676 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.84437966496191401156316829978, −11.82069309479297073675928774214, −11.13165633020076596959326662164, −10.94787131980350351962275402042, −10.63185389046844427398767225430, −9.941312811394734616827437330305, −8.449848652448631042664152728086, −8.413377543717375617545702426836, −7.57124551990048567275701191475, −7.06185511834118293178324786866, −6.12136274011557711041809849228, −6.07628190405153258233047025570, −5.03963424363305903155129842208, −4.88946697366976593865968375878, −4.34109928090453165595417566774, −3.56905866703009578913117371452, −2.36734827443036508490741970084, −1.74168741956282258555072528413, 0, 0,
1.74168741956282258555072528413, 2.36734827443036508490741970084, 3.56905866703009578913117371452, 4.34109928090453165595417566774, 4.88946697366976593865968375878, 5.03963424363305903155129842208, 6.07628190405153258233047025570, 6.12136274011557711041809849228, 7.06185511834118293178324786866, 7.57124551990048567275701191475, 8.413377543717375617545702426836, 8.449848652448631042664152728086, 9.941312811394734616827437330305, 10.63185389046844427398767225430, 10.94787131980350351962275402042, 11.13165633020076596959326662164, 11.82069309479297073675928774214, 11.84437966496191401156316829978
