| L(s) = 1 | − 5-s + 2·9-s − 2·13-s + 2·17-s − 4·25-s + 8·29-s − 2·37-s + 6·41-s − 2·45-s − 8·49-s + 4·53-s − 14·61-s + 2·65-s + 14·73-s − 5·81-s − 2·85-s − 6·97-s + 28·101-s − 10·109-s − 2·113-s − 4·117-s + 10·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 2/3·9-s − 0.554·13-s + 0.485·17-s − 4/5·25-s + 1.48·29-s − 0.328·37-s + 0.937·41-s − 0.298·45-s − 8/7·49-s + 0.549·53-s − 1.79·61-s + 0.248·65-s + 1.63·73-s − 5/9·81-s − 0.216·85-s − 0.609·97-s + 2.78·101-s − 0.957·109-s − 0.188·113-s − 0.369·117-s + 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 801280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 801280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.694627350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.694627350\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 313 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 22 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.119634481305738071123399420192, −7.85396510444853224741609668626, −7.35398494320109052899627442695, −7.03963221899575214021011360344, −6.41918780197521461852547701429, −6.13382393747150566759214516686, −5.44760467287847334909932701365, −5.02517545405833787481938675174, −4.37822967259721271041578346582, −4.23606825577748910066616490580, −3.40177809391979676567495303979, −3.02526961246026800201834205106, −2.25042879335001707643794410697, −1.57279490093297699153723723336, −0.64568414196424809713105122281,
0.64568414196424809713105122281, 1.57279490093297699153723723336, 2.25042879335001707643794410697, 3.02526961246026800201834205106, 3.40177809391979676567495303979, 4.23606825577748910066616490580, 4.37822967259721271041578346582, 5.02517545405833787481938675174, 5.44760467287847334909932701365, 6.13382393747150566759214516686, 6.41918780197521461852547701429, 7.03963221899575214021011360344, 7.35398494320109052899627442695, 7.85396510444853224741609668626, 8.119634481305738071123399420192
