| L(s) = 1 | − 32·2-s − 14·3-s + 768·4-s − 2.73e3·5-s + 448·6-s + 4.80e3·7-s − 1.63e4·8-s + 1.84e4·9-s + 8.73e4·10-s + 4.49e4·11-s − 1.07e4·12-s + 1.00e5·13-s − 1.53e5·14-s + 3.82e4·15-s + 3.27e5·16-s − 8.70e5·17-s − 5.88e5·18-s + 5.08e5·19-s − 2.09e6·20-s − 6.72e4·21-s − 1.43e6·22-s + 7.98e4·23-s + 2.29e5·24-s + 2.69e6·25-s − 3.20e6·26-s − 7.88e5·27-s + 3.68e6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.0997·3-s + 3/2·4-s − 1.95·5-s + 0.141·6-s + 0.755·7-s − 1.41·8-s + 0.935·9-s + 2.76·10-s + 0.925·11-s − 0.149·12-s + 0.973·13-s − 1.06·14-s + 0.194·15-s + 5/4·16-s − 2.52·17-s − 1.32·18-s + 0.895·19-s − 2.93·20-s − 0.0754·21-s − 1.30·22-s + 0.0594·23-s + 0.141·24-s + 1.38·25-s − 1.37·26-s − 0.285·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6679190582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6679190582\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 14 T - 6070 p T^{2} + 14 p^{9} T^{3} + p^{18} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 546 p T + 950594 p T^{2} + 546 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 44940 T + 2679533782 T^{2} - 44940 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7714 p T + 23606428002 T^{2} - 7714 p^{10} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 870408 T + 420751004110 T^{2} + 870408 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 508774 T + 15410410458 p T^{2} - 508774 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 79800 T + 3486389354926 T^{2} - 79800 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2006328 T - 2889837567866 T^{2} - 2006328 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2188732 T + 50511394548798 T^{2} - 2188732 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 20723576 T + 320148592616598 T^{2} + 20723576 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 19016592 T + 126947391521038 T^{2} - 19016592 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4193716 T + 733843976191350 T^{2} - 4193716 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 74542524 T + 3534811031083678 T^{2} + 74542524 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3239748 T + 6104780971308142 T^{2} + 3239748 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 133642362 T + 18901037953676014 T^{2} + 133642362 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 227801686 T + 34632227131611306 T^{2} - 227801686 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 332930272 T + 75814987106046390 T^{2} - 332930272 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 167985720 T + 6569741497979662 T^{2} + 167985720 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 44684276 T + 34064802404568870 T^{2} + 44684276 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 269642776 T + 246583195950001182 T^{2} - 269642776 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 183105762 T + 297182791992067342 T^{2} + 183105762 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 791657748 T + 751278555124390294 T^{2} - 791657748 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4169480 T + 1069351625837487534 T^{2} + 4169480 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−17.64358468950992880118329828692, −17.48318351881294072084276804829, −16.11390380300223324573779282756, −15.86334017545217412238557025563, −15.54317248974775814168880334356, −14.73286225778781967915854415218, −13.57753063940629120551869376986, −12.46940879253967871053670031408, −11.53283941259280985314340398088, −11.40144586360516622811431535984, −10.69304700240158331944415587905, −9.483992854131210120570944803065, −8.589346429552756975371994559532, −8.135771201621501665754483740013, −7.16854073133309675418502732780, −6.61000125644839319569772439103, −4.53750457018326753286737202859, −3.66154901038404841063752912294, −1.72061557408656683350454753851, −0.58290378085129920473216583401,
0.58290378085129920473216583401, 1.72061557408656683350454753851, 3.66154901038404841063752912294, 4.53750457018326753286737202859, 6.61000125644839319569772439103, 7.16854073133309675418502732780, 8.135771201621501665754483740013, 8.589346429552756975371994559532, 9.483992854131210120570944803065, 10.69304700240158331944415587905, 11.40144586360516622811431535984, 11.53283941259280985314340398088, 12.46940879253967871053670031408, 13.57753063940629120551869376986, 14.73286225778781967915854415218, 15.54317248974775814168880334356, 15.86334017545217412238557025563, 16.11390380300223324573779282756, 17.48318351881294072084276804829, 17.64358468950992880118329828692
