| L(s) = 1 | − 7·2-s − 6·3-s + 25·4-s − 10·5-s + 42·6-s − 14·7-s − 63·8-s + 27·9-s + 70·10-s − 26·11-s − 150·12-s + 14·13-s + 98·14-s + 60·15-s + 169·16-s + 16·17-s − 189·18-s + 174·19-s − 250·20-s + 84·21-s + 182·22-s + 184·23-s + 378·24-s + 75·25-s − 98·26-s − 108·27-s − 350·28-s + ⋯ |
| L(s) = 1 | − 2.47·2-s − 1.15·3-s + 25/8·4-s − 0.894·5-s + 2.85·6-s − 0.755·7-s − 2.78·8-s + 9-s + 2.21·10-s − 0.712·11-s − 3.60·12-s + 0.298·13-s + 1.87·14-s + 1.03·15-s + 2.64·16-s + 0.228·17-s − 2.47·18-s + 2.10·19-s − 2.79·20-s + 0.872·21-s + 1.76·22-s + 1.66·23-s + 3.21·24-s + 3/5·25-s − 0.739·26-s − 0.769·27-s − 2.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2818943105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2818943105\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 7 T + 3 p^{3} T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 26 T + 2406 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 14 T + 2386 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 16 T + 6558 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 174 T + 18414 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 p T + 19470 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 32 T + 19046 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 330 T + 85430 T^{2} - 330 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 132 T + 103214 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 200 T + 64542 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 364 T + 172486 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 292 T + 179390 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 34 T + 103410 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 364 T + 438374 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 792 T + 340886 T^{2} - 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 788 T + 753430 T^{2} + 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 454 T + 304526 T^{2} - 454 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 778 T + 794698 T^{2} - 778 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 408 T + 994782 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1136 T + 1169990 T^{2} - 1136 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 36 T - 842170 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 498 T + 1796754 T^{2} + 498 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.31646125251727793414102951075, −12.73463842502283315448535729326, −12.13888275203938453670504022247, −11.80488614034824292984902433281, −11.09249947415442609094036473962, −10.85238919864586517760511725224, −10.03660812578683913623067715150, −10.01769413808311710319666896620, −9.065205903528478871780054711906, −8.985366466559564523501598331861, −7.900290090280939225283356897271, −7.71569371610769471108144684916, −7.10252486838388399925964054620, −6.53826844183032639469487754512, −5.60465407862474500342670745560, −5.05609022035684486305391121950, −3.71172028367639825515238018771, −2.80644565124968087718140180505, −0.874669295656779778083072113373, −0.73492990724628868984135827920,
0.73492990724628868984135827920, 0.874669295656779778083072113373, 2.80644565124968087718140180505, 3.71172028367639825515238018771, 5.05609022035684486305391121950, 5.60465407862474500342670745560, 6.53826844183032639469487754512, 7.10252486838388399925964054620, 7.71569371610769471108144684916, 7.900290090280939225283356897271, 8.985366466559564523501598331861, 9.065205903528478871780054711906, 10.01769413808311710319666896620, 10.03660812578683913623067715150, 10.85238919864586517760511725224, 11.09249947415442609094036473962, 11.80488614034824292984902433281, 12.13888275203938453670504022247, 12.73463842502283315448535729326, 13.31646125251727793414102951075
