| L(s) = 1 | − 4·4-s − 2·9-s + 12·11-s + 4·16-s + 14·19-s − 6·29-s − 28·31-s + 8·36-s + 12·41-s − 48·44-s − 4·49-s − 30·59-s − 10·61-s + 16·64-s + 6·71-s − 56·76-s + 10·79-s + 9·81-s − 30·89-s − 24·99-s − 30·101-s + 22·109-s + 24·116-s + 46·121-s + 112·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s − 2/3·9-s + 3.61·11-s + 16-s + 3.21·19-s − 1.11·29-s − 5.02·31-s + 4/3·36-s + 1.87·41-s − 7.23·44-s − 4/7·49-s − 3.90·59-s − 1.28·61-s + 2·64-s + 0.712·71-s − 6.42·76-s + 1.12·79-s + 81-s − 3.17·89-s − 2.41·99-s − 2.98·101-s + 2.10·109-s + 2.22·116-s + 4.18·121-s + 10.0·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8203643045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8203643045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 82 T^{2} + 1395 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.030173661203425597557555437492, −7.77464208771795972208962237532, −7.47027233913951022615622415120, −7.32686579733638003288005296332, −6.81146538933171466891201736027, −6.78498553458026951830260153804, −6.76772306089893675247472780653, −5.96004480878040017705807850892, −5.86974380884192101539714078010, −5.68195054417327968144985517669, −5.65087247543136885852418416393, −4.94418267041729288861672119069, −4.93651899916904404292786676175, −4.81245366832967215775226340318, −4.14581044886372423467508100544, −3.88816202389305248278849273958, −3.88766642861698776903591404347, −3.73746216017072337283333209722, −3.18598208237196360298998038068, −3.14190017737766234071291479024, −2.46650914425931066872841915969, −1.68963123435614751517504654443, −1.51400027269807577686427754073, −1.26052134014176786655065684227, −0.35725981704408066408398955194,
0.35725981704408066408398955194, 1.26052134014176786655065684227, 1.51400027269807577686427754073, 1.68963123435614751517504654443, 2.46650914425931066872841915969, 3.14190017737766234071291479024, 3.18598208237196360298998038068, 3.73746216017072337283333209722, 3.88766642861698776903591404347, 3.88816202389305248278849273958, 4.14581044886372423467508100544, 4.81245366832967215775226340318, 4.93651899916904404292786676175, 4.94418267041729288861672119069, 5.65087247543136885852418416393, 5.68195054417327968144985517669, 5.86974380884192101539714078010, 5.96004480878040017705807850892, 6.76772306089893675247472780653, 6.78498553458026951830260153804, 6.81146538933171466891201736027, 7.32686579733638003288005296332, 7.47027233913951022615622415120, 7.77464208771795972208962237532, 8.030173661203425597557555437492
