| L(s) = 1 | − 2·2-s + 9·3-s + 16·4-s + 18·5-s − 18·6-s + 77·7-s − 88·8-s + 54·9-s − 36·10-s − 194·11-s + 144·12-s − 154·14-s + 162·15-s + 176·16-s − 420·17-s − 108·18-s + 453·19-s + 288·20-s + 693·21-s + 388·22-s + 112·23-s − 792·24-s − 409·25-s + 243·27-s + 1.23e3·28-s + 2.08e3·29-s − 324·30-s + ⋯ |
| L(s) = 1 | − 1/2·2-s + 3-s + 4-s + 0.719·5-s − 1/2·6-s + 11/7·7-s − 1.37·8-s + 2/3·9-s − 0.359·10-s − 1.60·11-s + 12-s − 0.785·14-s + 0.719·15-s + 0.687·16-s − 1.45·17-s − 1/3·18-s + 1.25·19-s + 0.719·20-s + 11/7·21-s + 0.801·22-s + 0.211·23-s − 1.37·24-s − 0.654·25-s + 1/3·27-s + 11/7·28-s + 2.47·29-s − 0.359·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.051814526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051814526\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 7 | $C_2$ | \( 1 - 11 p T + p^{4} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T - 3 p^{2} T^{2} + p^{5} T^{3} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 18 T + 733 T^{2} - 18 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 194 T + 22995 T^{2} + 194 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 30047 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 420 T + 142321 T^{2} + 420 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 453 T + 198724 T^{2} - 453 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 112 T - 267297 T^{2} - 112 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1040 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2019 T + 2282308 T^{2} + 2019 p^{4} T^{3} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1075 T - 718536 T^{2} - 1075 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3945974 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1087 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3750 T + 9567181 T^{2} - 3750 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2200 T - 3050481 T^{2} - 2200 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9264 T + 40724593 T^{2} + 9264 p^{4} T^{3} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 1212 T + 14335489 T^{2} - 1212 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2375 T - 14510496 T^{2} + 2375 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8938 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 15807 T + 111685324 T^{2} - 15807 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8147 T + 27423528 T^{2} + 8147 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50356694 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 23628 T + 248836369 T^{2} - 23628 p^{4} T^{3} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 164817362 T^{2} + p^{8} T^{4} \) |
| 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
−17.78340213118514644992673709190, −17.46974214424617603279787188109, −16.20309404186310495517491317802, −15.77343176117851996270227029567, −15.16541431360247892732185662370, −14.71232519218962986372349419454, −13.75806710007529918113110688400, −13.50148474438913402447037482135, −12.38183352896072947858929958054, −11.62602351913222686438257381884, −10.86756305348377399011198434557, −10.32117975401905520903699440027, −9.240818625419382141274034358616, −8.687032698455008374623286530672, −7.83594857995684727959195736051, −7.23153127298501343977157287942, −5.93617298363017033538052244961, −4.86556202543996125709901766866, −2.81921384214715572436655270343, −1.97587835613929894864156584210,
1.97587835613929894864156584210, 2.81921384214715572436655270343, 4.86556202543996125709901766866, 5.93617298363017033538052244961, 7.23153127298501343977157287942, 7.83594857995684727959195736051, 8.687032698455008374623286530672, 9.240818625419382141274034358616, 10.32117975401905520903699440027, 10.86756305348377399011198434557, 11.62602351913222686438257381884, 12.38183352896072947858929958054, 13.50148474438913402447037482135, 13.75806710007529918113110688400, 14.71232519218962986372349419454, 15.16541431360247892732185662370, 15.77343176117851996270227029567, 16.20309404186310495517491317802, 17.46974214424617603279787188109, 17.78340213118514644992673709190
