| L(s) = 1 | + 3·4-s + 2·11-s + 5·16-s + 10·29-s − 2·31-s − 12·41-s + 6·44-s + 13·49-s − 6·59-s − 14·61-s + 3·64-s + 24·79-s + 16·89-s + 6·101-s + 4·109-s + 30·116-s − 19·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 36·164-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 0.603·11-s + 5/4·16-s + 1.85·29-s − 0.359·31-s − 1.87·41-s + 0.904·44-s + 13/7·49-s − 0.781·59-s − 1.79·61-s + 3/8·64-s + 2.70·79-s + 1.69·89-s + 0.597·101-s + 0.383·109-s + 2.78·116-s − 1.72·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 2.81·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.341080236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.341080236\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 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
−8.768564274085087932831364992412, −8.697229528157422290814209093098, −8.089774687373515274108627846159, −7.67676564255793711196944516647, −7.50350786769293033407815774572, −6.98301762308350469794994489940, −6.58645375874665331619468650552, −6.34278642132725592404018360353, −6.22495337733159156943557916783, −5.35947116048447598372575627148, −5.33980890076554215880493760359, −4.54058577903198037449659746514, −4.36440468665874198056588859072, −3.50444374115243041701506705708, −3.38992793783653184481310565357, −2.79196192908017103323774451460, −2.33920521698581285417961900230, −1.84770158407155639127832016898, −1.35900907194197540459043385723, −0.65507990564981916517891676902,
0.65507990564981916517891676902, 1.35900907194197540459043385723, 1.84770158407155639127832016898, 2.33920521698581285417961900230, 2.79196192908017103323774451460, 3.38992793783653184481310565357, 3.50444374115243041701506705708, 4.36440468665874198056588859072, 4.54058577903198037449659746514, 5.33980890076554215880493760359, 5.35947116048447598372575627148, 6.22495337733159156943557916783, 6.34278642132725592404018360353, 6.58645375874665331619468650552, 6.98301762308350469794994489940, 7.50350786769293033407815774572, 7.67676564255793711196944516647, 8.089774687373515274108627846159, 8.697229528157422290814209093098, 8.768564274085087932831364992412
