| L(s) = 1 | + 2-s − 2·3-s − 2·4-s − 2·6-s + 7-s − 3·8-s − 3·9-s − 6·11-s + 4·12-s + 3·13-s + 14-s + 16-s + 6·17-s − 3·18-s − 5·19-s − 2·21-s − 6·22-s − 12·23-s + 6·24-s + 3·26-s + 14·27-s − 2·28-s − 5·29-s − 6·31-s + 2·32-s + 12·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 4-s − 0.816·6-s + 0.377·7-s − 1.06·8-s − 9-s − 1.80·11-s + 1.15·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.14·19-s − 0.436·21-s − 1.27·22-s − 2.50·23-s + 1.22·24-s + 0.588·26-s + 2.69·27-s − 0.377·28-s − 0.928·29-s − 1.07·31-s + 0.353·32-s + 2.08·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 163 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 171 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 133 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 177 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 183 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−10.37836332430107856791355928771, −10.33974219662881949813925812997, −9.472106703208809998437932716325, −9.158769433767514207650242644238, −8.402407948670096481927558824071, −8.271859739401738772609051754587, −7.911494512284177911245429121248, −7.37965171227660629349863866049, −6.39846032466690188750117531412, −6.03946997729463624803435756299, −5.61936346762485966861314235998, −5.50919505674424159480558476977, −4.82603976515769184310932348181, −4.63158370589134938107945374811, −3.61982368548635112315187805330, −3.49313837703386866870810729024, −2.58187502389972903250512989258, −1.70627417996651129056379359719, 0, 0,
1.70627417996651129056379359719, 2.58187502389972903250512989258, 3.49313837703386866870810729024, 3.61982368548635112315187805330, 4.63158370589134938107945374811, 4.82603976515769184310932348181, 5.50919505674424159480558476977, 5.61936346762485966861314235998, 6.03946997729463624803435756299, 6.39846032466690188750117531412, 7.37965171227660629349863866049, 7.911494512284177911245429121248, 8.271859739401738772609051754587, 8.402407948670096481927558824071, 9.158769433767514207650242644238, 9.472106703208809998437932716325, 10.33974219662881949813925812997, 10.37836332430107856791355928771
