| L(s) = 1 | − 2·2-s + 2·4-s − 12·5-s − 10·7-s − 4·8-s − 3·9-s + 24·10-s + 12·11-s + 20·14-s + 8·16-s + 24·17-s + 6·18-s + 62·19-s − 24·20-s − 24·22-s + 48·23-s + 72·25-s − 20·28-s + 106·31-s − 8·32-s − 48·34-s + 120·35-s − 6·36-s − 98·37-s − 124·38-s + 48·40-s − 96·41-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 2.39·5-s − 1.42·7-s − 1/2·8-s − 1/3·9-s + 12/5·10-s + 1.09·11-s + 10/7·14-s + 1/2·16-s + 1.41·17-s + 1/3·18-s + 3.26·19-s − 6/5·20-s − 1.09·22-s + 2.08·23-s + 2.87·25-s − 5/7·28-s + 3.41·31-s − 1/4·32-s − 1.41·34-s + 24/7·35-s − 1/6·36-s − 2.64·37-s − 3.26·38-s + 6/5·40-s − 2.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6479027952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6479027952\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2594 T^{4} + 444 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 10 T + 125 T^{2} + 990 T^{3} + 8024 T^{4} + 990 p^{2} T^{5} + 125 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T + 180 T^{2} + 612 T^{3} - 1921 T^{4} + 612 p^{2} T^{5} + 180 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 24 T + 674 T^{2} - 11568 T^{3} + 204291 T^{4} - 11568 p^{2} T^{5} + 674 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 62 T + 986 T^{2} + 19992 T^{3} - 879313 T^{4} + 19992 p^{2} T^{5} + 986 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 48 T + 1694 T^{2} - 44448 T^{3} + 983907 T^{4} - 44448 p^{2} T^{5} + 1694 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 1382 T^{2} + 1202643 T^{4} - 1382 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 106 T + 5618 T^{2} - 246768 T^{3} + 8970479 T^{4} - 246768 p^{2} T^{5} + 5618 p^{4} T^{6} - 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 98 T + 5882 T^{2} + 250872 T^{3} + 9855887 T^{4} + 250872 p^{2} T^{5} + 5882 p^{4} T^{6} + 98 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 96 T + 3600 T^{2} + 39144 T^{3} - 1284193 T^{4} + 39144 p^{2} T^{5} + 3600 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 30 T + 3497 T^{2} - 95910 T^{3} + 7356708 T^{4} - 95910 p^{2} T^{5} + 3497 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T + 8712 T^{2} - 400884 T^{3} + 17761154 T^{4} - 400884 p^{2} T^{5} + 8712 p^{4} T^{6} - 132 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 36 T - 406 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 84 T + 4068 T^{2} + 238356 T^{3} - 20158273 T^{4} + 238356 p^{2} T^{5} + 4068 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 72 T - 1679 T^{2} - 41688 T^{3} + 13965264 T^{4} - 41688 p^{2} T^{5} - 1679 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 148 T + 8501 T^{2} + 114648 T^{3} - 12152596 T^{4} + 114648 p^{2} T^{5} + 8501 p^{4} T^{6} + 148 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 180 T + 8676 T^{2} + 872292 T^{3} - 126145969 T^{4} + 872292 p^{2} T^{5} + 8676 p^{4} T^{6} - 180 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 190 T + 18050 T^{2} + 1719120 T^{3} + 149901647 T^{4} + 1719120 p^{2} T^{5} + 18050 p^{4} T^{6} + 190 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 48 T + 12911 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 264 T + 34848 T^{2} - 3047880 T^{3} + 244895714 T^{4} - 3047880 p^{2} T^{5} + 34848 p^{4} T^{6} - 264 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 288 T + 21312 T^{2} + 2099280 T^{3} - 430487521 T^{4} + 2099280 p^{2} T^{5} + 21312 p^{4} T^{6} - 288 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 310 T + 27389 T^{2} + 1493934 T^{3} - 415402504 T^{4} + 1493934 p^{2} T^{5} + 27389 p^{4} T^{6} - 310 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.51641385646413912354650907864, −10.12362367454558105625979606754, −9.798512339003546038795027891924, −9.643585402789329734612986007535, −9.088788685155377069845353284056, −8.954243797068837779020476594950, −8.791624177825614816892816836658, −8.429120050449098872150162202723, −7.907370579766661779014210040975, −7.66779612081766396050643280685, −7.54636318421128527369071598232, −7.14017848876503375876394850936, −6.76798359684675156517611668642, −6.64969306792867401484932388048, −6.15013208642673797305223854709, −5.58411985960182780956895750200, −5.08331229238324740830885555784, −4.98287240467351646371301914262, −4.28135235419446707613103269270, −3.46449526572587331397000264109, −3.45766718346058524052313076119, −3.30530792085793444300787954630, −2.77955414022913292579963359124, −0.999668568583824310114476324065, −0.78073817612491168013945308798,
0.78073817612491168013945308798, 0.999668568583824310114476324065, 2.77955414022913292579963359124, 3.30530792085793444300787954630, 3.45766718346058524052313076119, 3.46449526572587331397000264109, 4.28135235419446707613103269270, 4.98287240467351646371301914262, 5.08331229238324740830885555784, 5.58411985960182780956895750200, 6.15013208642673797305223854709, 6.64969306792867401484932388048, 6.76798359684675156517611668642, 7.14017848876503375876394850936, 7.54636318421128527369071598232, 7.66779612081766396050643280685, 7.907370579766661779014210040975, 8.429120050449098872150162202723, 8.791624177825614816892816836658, 8.954243797068837779020476594950, 9.088788685155377069845353284056, 9.643585402789329734612986007535, 9.798512339003546038795027891924, 10.12362367454558105625979606754, 10.51641385646413912354650907864
