| L(s) = 1 | + 42·3-s − 14·5-s + 686·7-s − 810·9-s + 540·11-s + 462·13-s − 588·15-s − 9.12e3·17-s + 5.26e4·19-s + 2.88e4·21-s − 1.15e5·23-s − 1.25e5·25-s − 5.02e4·27-s + 6.86e4·29-s − 2.35e5·31-s + 2.26e4·33-s − 9.60e3·35-s + 8.74e4·37-s + 1.94e4·39-s + 2.15e5·41-s + 3.50e5·43-s + 1.13e4·45-s − 7.73e5·47-s + 3.52e5·49-s − 3.83e5·51-s − 5.94e5·53-s − 7.56e3·55-s + ⋯ |
| L(s) = 1 | + 0.898·3-s − 0.0500·5-s + 0.755·7-s − 0.370·9-s + 0.122·11-s + 0.0583·13-s − 0.0449·15-s − 0.450·17-s + 1.76·19-s + 0.678·21-s − 1.98·23-s − 1.61·25-s − 0.491·27-s + 0.522·29-s − 1.41·31-s + 0.109·33-s − 0.0378·35-s + 0.283·37-s + 0.0523·39-s + 0.487·41-s + 0.672·43-s + 0.0185·45-s − 1.08·47-s + 3/7·49-s − 0.404·51-s − 0.548·53-s − 0.00612·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 14 p T + 286 p^{2} T^{2} - 14 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 14 T + 25234 p T^{2} + 14 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 540 T + 38608006 T^{2} - 540 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 462 T + 83897426 T^{2} - 462 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9128 T + 833968718 T^{2} + 9128 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 52654 T + 1658783646 T^{2} - 52654 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 115864 T + 8328980174 T^{2} + 115864 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 68680 T + 29105626262 T^{2} - 68680 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7588 p T + 49338345534 T^{2} + 7588 p^{8} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 87480 T + 173197795910 T^{2} - 87480 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 215152 T + 387453272942 T^{2} - 215152 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 350372 T + 452955273894 T^{2} - 350372 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 773668 T + 725865082718 T^{2} + 773668 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 594956 T + 32834798558 T^{2} + 594956 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 806274 T + 4060725990718 T^{2} + 806274 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3168830 T + 7810177890906 T^{2} + 3168830 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2815488 T + 11472388517126 T^{2} - 2815488 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 754824 T + 10389020745070 T^{2} + 754824 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3907820 T + 22920441770694 T^{2} - 3907820 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5878360 T + 23857239598302 T^{2} + 5878360 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2348346 T + 33635389849342 T^{2} + 2348346 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8096396 T + 37284598214006 T^{2} + 8096396 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18166120 T + 222816303452142 T^{2} + 18166120 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−9.527924815339979025263510057128, −9.420850816779984526063477179153, −8.685799079446615291091666663137, −8.357264657295481556395139323056, −7.83496484038830031850656995848, −7.67286665370936358817447986256, −7.18397951309457562776671464735, −6.38826336286782692596474501709, −5.83952486659120305080948113973, −5.61404892767704206772143130248, −4.87122388657257849121867962806, −4.37781883405165493663106518607, −3.65892310046599849343009836481, −3.51039968371510075211036954071, −2.58614121186491720094226173676, −2.33700578024371851188507588277, −1.54831940492250953697221899594, −1.24438801780527054263464401357, 0, 0,
1.24438801780527054263464401357, 1.54831940492250953697221899594, 2.33700578024371851188507588277, 2.58614121186491720094226173676, 3.51039968371510075211036954071, 3.65892310046599849343009836481, 4.37781883405165493663106518607, 4.87122388657257849121867962806, 5.61404892767704206772143130248, 5.83952486659120305080948113973, 6.38826336286782692596474501709, 7.18397951309457562776671464735, 7.67286665370936358817447986256, 7.83496484038830031850656995848, 8.357264657295481556395139323056, 8.685799079446615291091666663137, 9.420850816779984526063477179153, 9.527924815339979025263510057128
