| L(s) = 1 | + 6·2-s − 42·3-s + 128·4-s − 84·5-s − 252·6-s + 2.08e3·8-s + 2.18e3·9-s − 504·10-s + 5.56e3·11-s − 5.37e3·12-s + 1.03e4·13-s + 3.52e3·15-s + 1.25e4·16-s − 1.39e4·17-s + 1.31e4·18-s + 5.53e4·19-s − 1.07e4·20-s + 3.34e4·22-s + 9.12e4·23-s − 8.76e4·24-s + 7.81e4·25-s + 6.18e4·26-s − 2.01e5·27-s + 8.32e4·29-s + 2.11e4·30-s + 1.50e5·31-s + 2.67e5·32-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 0.898·3-s + 4-s − 0.300·5-s − 0.476·6-s + 1.44·8-s + 9-s − 0.159·10-s + 1.26·11-s − 0.898·12-s + 1.30·13-s + 0.269·15-s + 0.764·16-s − 0.690·17-s + 0.530·18-s + 1.85·19-s − 0.300·20-s + 0.668·22-s + 1.56·23-s − 1.29·24-s + 25-s + 0.689·26-s − 1.96·27-s + 0.633·29-s + 0.143·30-s + 0.906·31-s + 1.44·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.868217338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.868217338\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p T - 23 p^{2} T^{2} - 3 p^{8} T^{3} + p^{14} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 14 p T - 47 p^{2} T^{2} + 14 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 84 T - 71069 T^{2} + 84 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5568 T + 11515453 T^{2} - 5568 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5152 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 13986 T - 214730477 T^{2} + 13986 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 55370 T + 2171965161 T^{2} - 55370 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 91272 T + 4925752537 T^{2} - 91272 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 41610 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 150332 T - 4912903887 T^{2} - 150332 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 136366 T - 76336191177 T^{2} - 136366 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 510258 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 172072 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 519036 T - 237224751167 T^{2} + 519036 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 59202 T - 1171206263033 T^{2} - 59202 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1979250 T + 1428779077681 T^{2} - 1979250 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2988748 T + 5789871771483 T^{2} + 2988748 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2409404 T - 255483970107 T^{2} + 2409404 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1504512 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1821022 T - 7731277394613 T^{2} + 1821022 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1669240 T - 16417546808559 T^{2} - 1669240 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 696738 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 5558490 T - 13334523815429 T^{2} - 5558490 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 101822 p T + p^{7} T^{2} )^{2} \) |
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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
−14.40591685372181726399045825316, −13.53803198236763838544777526103, −13.40498860176904166377142318774, −12.63366814638080064746727137989, −11.78696403231681380475887039981, −11.60556542668725723816973507553, −10.90035893914641952164294288928, −10.77041974407164282372210092802, −9.651022714812939654547548317915, −9.114069267933346319654723565261, −8.016281750306758096755332324395, −7.29206463168736336374377129753, −6.72462174503420333161805255610, −6.27571594301294274646563147979, −5.32373901489128142115944817965, −4.52047436501851343676281773259, −3.87434579094005729480139856716, −2.81702066467719287029184321868, −1.23871927997160520166034294244, −1.14327375036693190033429883802,
1.14327375036693190033429883802, 1.23871927997160520166034294244, 2.81702066467719287029184321868, 3.87434579094005729480139856716, 4.52047436501851343676281773259, 5.32373901489128142115944817965, 6.27571594301294274646563147979, 6.72462174503420333161805255610, 7.29206463168736336374377129753, 8.016281750306758096755332324395, 9.114069267933346319654723565261, 9.651022714812939654547548317915, 10.77041974407164282372210092802, 10.90035893914641952164294288928, 11.60556542668725723816973507553, 11.78696403231681380475887039981, 12.63366814638080064746727137989, 13.40498860176904166377142318774, 13.53803198236763838544777526103, 14.40591685372181726399045825316
