| L(s) = 1 | − 2·3-s + 4·5-s + 2·7-s + 2·9-s + 2·11-s − 8·15-s + 2·17-s − 4·19-s − 4·21-s + 2·23-s + 2·25-s − 6·27-s + 4·29-s − 2·31-s − 4·33-s + 8·35-s − 4·37-s + 4·41-s − 12·43-s + 8·45-s + 8·47-s − 6·49-s − 4·51-s − 4·53-s + 8·55-s + 8·57-s + 20·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 0.755·7-s + 2/3·9-s + 0.603·11-s − 2.06·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s + 0.417·23-s + 2/5·25-s − 1.15·27-s + 0.742·29-s − 0.359·31-s − 0.696·33-s + 1.35·35-s − 0.657·37-s + 0.624·41-s − 1.82·43-s + 1.19·45-s + 1.16·47-s − 6/7·49-s − 0.560·51-s − 0.549·53-s + 1.07·55-s + 1.05·57-s + 2.60·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.137059284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.137059284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 186 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 302 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−13.49085759577834040588555307572, −12.95499863335649123679834245047, −12.44836080142256483393647294834, −11.88176029212561145998688564212, −11.40819645886218929442505664158, −10.99546039656749745576780395701, −10.29935346640717734907909684785, −10.05473493622270074119771615729, −9.440210581016239559833347742153, −8.942534572435227892849471776987, −8.213281007835140825652083695990, −7.54227009460585271658092107954, −6.55803287402778873278660169255, −6.44153995685194095508342264679, −5.59734670055123138599998759290, −5.37445701147005362698510239113, −4.62453500205073604857732525057, −3.69329493216429982385966860300, −2.24827141653266557516415079165, −1.50989576196998004253019603835,
1.50989576196998004253019603835, 2.24827141653266557516415079165, 3.69329493216429982385966860300, 4.62453500205073604857732525057, 5.37445701147005362698510239113, 5.59734670055123138599998759290, 6.44153995685194095508342264679, 6.55803287402778873278660169255, 7.54227009460585271658092107954, 8.213281007835140825652083695990, 8.942534572435227892849471776987, 9.440210581016239559833347742153, 10.05473493622270074119771615729, 10.29935346640717734907909684785, 10.99546039656749745576780395701, 11.40819645886218929442505664158, 11.88176029212561145998688564212, 12.44836080142256483393647294834, 12.95499863335649123679834245047, 13.49085759577834040588555307572
