| L(s) = 1 | + 3·4-s − 4·11-s + 5·16-s + 16·19-s + 2·29-s + 10·41-s − 12·44-s + 5·49-s + 28·59-s + 14·61-s + 3·64-s + 4·71-s + 48·76-s + 12·79-s + 30·89-s − 36·101-s − 10·109-s + 6·116-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1.20·11-s + 5/4·16-s + 3.67·19-s + 0.371·29-s + 1.56·41-s − 1.80·44-s + 5/7·49-s + 3.64·59-s + 1.79·61-s + 3/8·64-s + 0.474·71-s + 5.50·76-s + 1.35·79-s + 3.17·89-s − 3.58·101-s − 0.957·109-s + 0.557·116-s − 0.909·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.584139565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.584139565\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
| 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
−9.262100517473119846947611803145, −9.216671908468601068603865209808, −8.246952852955722579449137054363, −8.142337194185448615514503784285, −7.64369990263612032540219959141, −7.43630468478314830095915477430, −7.01427774480794209098530282911, −6.74887922406124275868822580062, −6.25428291419858248444103050468, −5.48384601331890464584641640781, −5.43454586698286757171213198556, −5.30118154035997702690623498826, −4.52654511275193127997975982567, −3.70198749896840164476878621518, −3.51554734338535868413404096591, −2.87682520648639803868689590344, −2.42156515848049926800147770809, −2.26407020116329291878464938219, −1.12932340481397501055521220643, −0.898631429461865569707751739729,
0.898631429461865569707751739729, 1.12932340481397501055521220643, 2.26407020116329291878464938219, 2.42156515848049926800147770809, 2.87682520648639803868689590344, 3.51554734338535868413404096591, 3.70198749896840164476878621518, 4.52654511275193127997975982567, 5.30118154035997702690623498826, 5.43454586698286757171213198556, 5.48384601331890464584641640781, 6.25428291419858248444103050468, 6.74887922406124275868822580062, 7.01427774480794209098530282911, 7.43630468478314830095915477430, 7.64369990263612032540219959141, 8.142337194185448615514503784285, 8.246952852955722579449137054363, 9.216671908468601068603865209808, 9.262100517473119846947611803145
