| L(s) = 1 | + 3-s − 4·4-s − 5-s − 2·9-s − 5·11-s − 4·12-s − 13-s − 15-s + 12·16-s + 3·17-s + 4·20-s − 4·23-s − 6·25-s − 2·27-s + 4·29-s − 31-s − 5·33-s + 8·36-s − 39-s − 2·41-s − 12·43-s + 20·44-s + 2·45-s − 18·47-s + 12·48-s + 3·51-s + 4·52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2·4-s − 0.447·5-s − 2/3·9-s − 1.50·11-s − 1.15·12-s − 0.277·13-s − 0.258·15-s + 3·16-s + 0.727·17-s + 0.894·20-s − 0.834·23-s − 6/5·25-s − 0.384·27-s + 0.742·29-s − 0.179·31-s − 0.870·33-s + 4/3·36-s − 0.160·39-s − 0.312·41-s − 1.82·43-s + 3.01·44-s + 0.298·45-s − 2.62·47-s + 1.73·48-s + 0.420·51-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 49 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 59 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_4$ | \( 1 - 9 T + 97 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 121 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 15 T + 199 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T - 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 217 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 181 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 247 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.636590382527388059495705961148, −8.538001658411728133882102327904, −8.307452332217837551766221746780, −8.022659963443799920566429253040, −7.55429520285359488801134062400, −7.39066871440018370840759796308, −6.38389042444821376318711734752, −6.21780991474801783122677631253, −5.39534440743052704069730544940, −5.34153929778258421376377261193, −4.91762191538086485946225308253, −4.58594813353187522678526475918, −3.74686760938035619442825419960, −3.71255051281043156486844540891, −3.20961502638969703775724182112, −2.65108000302143720807166783481, −2.01086183145411222651245440868, −1.11582280410594802144602502556, 0, 0,
1.11582280410594802144602502556, 2.01086183145411222651245440868, 2.65108000302143720807166783481, 3.20961502638969703775724182112, 3.71255051281043156486844540891, 3.74686760938035619442825419960, 4.58594813353187522678526475918, 4.91762191538086485946225308253, 5.34153929778258421376377261193, 5.39534440743052704069730544940, 6.21780991474801783122677631253, 6.38389042444821376318711734752, 7.39066871440018370840759796308, 7.55429520285359488801134062400, 8.022659963443799920566429253040, 8.307452332217837551766221746780, 8.538001658411728133882102327904, 8.636590382527388059495705961148
