| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 4·8-s + 4·10-s − 2·11-s − 10·13-s + 5·16-s + 12·17-s + 6·19-s + 6·20-s − 4·22-s + 2·23-s − 20·26-s − 2·29-s + 8·31-s + 6·32-s + 24·34-s + 2·37-s + 12·38-s + 8·40-s + 6·41-s − 8·43-s − 6·44-s + 4·46-s + 16·47-s − 30·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s − 0.603·11-s − 2.77·13-s + 5/4·16-s + 2.91·17-s + 1.37·19-s + 1.34·20-s − 0.852·22-s + 0.417·23-s − 3.92·26-s − 0.371·29-s + 1.43·31-s + 1.06·32-s + 4.11·34-s + 0.328·37-s + 1.94·38-s + 1.26·40-s + 0.937·41-s − 1.21·43-s − 0.904·44-s + 0.589·46-s + 2.33·47-s − 4.16·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.56943005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.56943005\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 40 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 130 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 100 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 115 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 22 T + 287 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−7.57126992278292893754976062388, −7.42936605814683152705648170066, −7.30113592767518377874468114493, −6.90953797554558321376812518546, −6.15187726486640824680597671163, −6.06844365581207846678053575818, −5.62299250578144162888608006712, −5.48258864815507642787907927932, −5.13968927244974983056683892428, −4.84797752189328296370878286199, −4.41487810670351610105115296256, −4.27147663649331068352651564575, −3.37121175504728817472963863558, −3.17874961556046510490202253988, −2.89224669341342942079316870344, −2.68823030177219340408620936685, −1.92947958163796855151529363336, −1.87749448016227591463693766668, −0.986360983275491990916946305414, −0.66836887622525648986119042678,
0.66836887622525648986119042678, 0.986360983275491990916946305414, 1.87749448016227591463693766668, 1.92947958163796855151529363336, 2.68823030177219340408620936685, 2.89224669341342942079316870344, 3.17874961556046510490202253988, 3.37121175504728817472963863558, 4.27147663649331068352651564575, 4.41487810670351610105115296256, 4.84797752189328296370878286199, 5.13968927244974983056683892428, 5.48258864815507642787907927932, 5.62299250578144162888608006712, 6.06844365581207846678053575818, 6.15187726486640824680597671163, 6.90953797554558321376812518546, 7.30113592767518377874468114493, 7.42936605814683152705648170066, 7.57126992278292893754976062388
