| L(s) = 1 | + 18·3-s − 78·5-s + 98·7-s + 243·9-s + 370·11-s − 1.05e3·13-s − 1.40e3·15-s − 1.02e3·17-s − 280·19-s + 1.76e3·21-s − 1.93e3·23-s + 3.13e3·25-s + 2.91e3·27-s + 2.55e3·29-s − 9.40e3·31-s + 6.66e3·33-s − 7.64e3·35-s − 168·37-s − 1.90e4·39-s − 8.16e3·41-s − 2.65e4·43-s − 1.89e4·45-s − 3.33e4·47-s + 7.20e3·49-s − 1.84e4·51-s + 3.59e4·53-s − 2.88e4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.39·5-s + 0.755·7-s + 9-s + 0.921·11-s − 1.73·13-s − 1.61·15-s − 0.861·17-s − 0.177·19-s + 0.872·21-s − 0.760·23-s + 1.00·25-s + 0.769·27-s + 0.564·29-s − 1.75·31-s + 1.06·33-s − 1.05·35-s − 0.0201·37-s − 2.00·39-s − 0.758·41-s − 2.18·43-s − 1.39·45-s − 2.20·47-s + 3/7·49-s − 0.994·51-s + 1.75·53-s − 1.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 78 T + 2946 T^{2} + 78 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 370 T + 92110 T^{2} - 370 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1056 T + 1002070 T^{2} + 1056 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1026 T + 3101146 T^{2} + 1026 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 280 T + 4527126 T^{2} + 280 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1930 T + 8676094 T^{2} + 1930 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2556 T + 42210910 T^{2} - 2556 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9408 T + 74445118 T^{2} + 9408 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 168 T - 25371242 T^{2} + 168 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8166 T + 110576466 T^{2} + 8166 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 26552 T + 467848070 T^{2} + 26552 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 33324 T + 728437086 T^{2} + 33324 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 35904 T + 1106075878 T^{2} - 35904 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 332 p T + 633628534 T^{2} - 332 p^{6} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 39028 T + 2066826686 T^{2} + 39028 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 57692 T + 3462397958 T^{2} + 57692 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9558 T + 1530801150 T^{2} + 9558 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8712 T - 1359024578 T^{2} - 8712 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 33420 T + 6193824670 T^{2} + 33420 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 17752 T + 7945747862 T^{2} + 17752 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 167718 T + 17824867954 T^{2} + 167718 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 46928 T + 163241630 p T^{2} + 46928 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.31685558446154178310715702040, −10.05495717819009732396194948904, −9.483832325207236552175598675228, −8.924945292991519994465890686364, −8.519059911660735179091722664641, −8.244158889013104205174064241731, −7.62157680970586809830829996555, −7.31815335625968647044186042610, −6.87796205081820817524132961498, −6.36910601865660901372724013143, −5.22695060429923522237596925266, −4.89892099513590356478160263040, −4.18150844105286706236335997196, −3.98640605820107238500241488748, −3.24985450308796590422261284450, −2.65931988772819559974016718986, −1.85106190556346480537127101507, −1.45650547824126596360619039640, 0, 0,
1.45650547824126596360619039640, 1.85106190556346480537127101507, 2.65931988772819559974016718986, 3.24985450308796590422261284450, 3.98640605820107238500241488748, 4.18150844105286706236335997196, 4.89892099513590356478160263040, 5.22695060429923522237596925266, 6.36910601865660901372724013143, 6.87796205081820817524132961498, 7.31815335625968647044186042610, 7.62157680970586809830829996555, 8.244158889013104205174064241731, 8.519059911660735179091722664641, 8.924945292991519994465890686364, 9.483832325207236552175598675228, 10.05495717819009732396194948904, 10.31685558446154178310715702040
