| L(s) = 1 | − 2·2-s + 6·3-s − 5·4-s − 10·5-s − 12·6-s + 12·8-s + 27·9-s + 20·10-s − 16·11-s − 30·12-s + 76·13-s − 60·15-s − 11·16-s + 124·17-s − 54·18-s + 96·19-s + 50·20-s + 32·22-s − 16·23-s + 72·24-s + 75·25-s − 152·26-s + 108·27-s + 188·29-s + 120·30-s + 120·31-s + 122·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 5/8·4-s − 0.894·5-s − 0.816·6-s + 0.530·8-s + 9-s + 0.632·10-s − 0.438·11-s − 0.721·12-s + 1.62·13-s − 1.03·15-s − 0.171·16-s + 1.76·17-s − 0.707·18-s + 1.15·19-s + 0.559·20-s + 0.310·22-s − 0.145·23-s + 0.612·24-s + 3/5·25-s − 1.14·26-s + 0.769·27-s + 1.20·29-s + 0.730·30-s + 0.695·31-s + 0.673·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.060445031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.060445031\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 9 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 16 T - 474 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 76 T + 5806 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 124 T + 13638 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 96 T + 13974 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 16 T + 15150 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 188 T + 34286 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 120 T + 60590 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 132 T + 70814 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 100 T + 89142 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 536 T + 200086 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 928 T + 408830 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 884 T + 460350 T^{2} - 884 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 104 T + 80534 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 468 T + 494606 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1688 T + 1302310 T^{2} + 1688 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 136 T + 540446 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 508 T + 13078 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 432 T + 602142 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 584 T + 1172390 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1404 T + 1802390 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1188 T + 2161254 T^{2} - 1188 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.04642680045731978221210064195, −9.888592815507846826580610397486, −8.976123561320397771592860616304, −8.888912747320645013370379942907, −8.587230537100032457373585557347, −8.234459399613759551139492024203, −7.62256471382080955531869526518, −7.55439853012923712801600649034, −6.99363800474875261165153274785, −6.29832077726096977510547032114, −5.72784522162088994712727506113, −5.23704571024188685083265849067, −4.54067852325523472305264939668, −4.18368211684997589417324072063, −3.38514782361171856859253523716, −3.37405971014763555008885041208, −2.71086674551446660179708001342, −1.72816161663802529250728229333, −0.851499036191040435296454975892, −0.76339598464648963152370000343,
0.76339598464648963152370000343, 0.851499036191040435296454975892, 1.72816161663802529250728229333, 2.71086674551446660179708001342, 3.37405971014763555008885041208, 3.38514782361171856859253523716, 4.18368211684997589417324072063, 4.54067852325523472305264939668, 5.23704571024188685083265849067, 5.72784522162088994712727506113, 6.29832077726096977510547032114, 6.99363800474875261165153274785, 7.55439853012923712801600649034, 7.62256471382080955531869526518, 8.234459399613759551139492024203, 8.587230537100032457373585557347, 8.888912747320645013370379942907, 8.976123561320397771592860616304, 9.888592815507846826580610397486, 10.04642680045731978221210064195
