| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 9-s − 3·10-s + 13-s + 16-s + 9·17-s + 18-s + 3·20-s + 4·25-s − 26-s − 12·29-s − 32-s − 9·34-s − 36-s − 14·37-s − 3·40-s − 18·41-s − 3·45-s − 4·49-s − 4·50-s + 52-s + 12·53-s + 12·58-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 1/3·9-s − 0.948·10-s + 0.277·13-s + 1/4·16-s + 2.18·17-s + 0.235·18-s + 0.670·20-s + 4/5·25-s − 0.196·26-s − 2.22·29-s − 0.176·32-s − 1.54·34-s − 1/6·36-s − 2.30·37-s − 0.474·40-s − 2.81·41-s − 0.447·45-s − 4/7·49-s − 0.565·50-s + 0.138·52-s + 1.64·53-s + 1.57·58-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 127 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 155 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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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
−7.989504075325736118469432593577, −7.24369508149471711275304229717, −6.98551012335732584811365619279, −6.65952288232795945572719434084, −5.80076760266709890201906162752, −5.70328401277451193613590378589, −5.34938426938765866033018577855, −4.97596463418536076827206608597, −3.84753328854586560015105894513, −3.55050615627217587603900701600, −3.03890029871320396274698942812, −2.30821015621813431381530578829, −1.61232372795041536992745023676, −1.38822650423888836568248197505, 0,
1.38822650423888836568248197505, 1.61232372795041536992745023676, 2.30821015621813431381530578829, 3.03890029871320396274698942812, 3.55050615627217587603900701600, 3.84753328854586560015105894513, 4.97596463418536076827206608597, 5.34938426938765866033018577855, 5.70328401277451193613590378589, 5.80076760266709890201906162752, 6.65952288232795945572719434084, 6.98551012335732584811365619279, 7.24369508149471711275304229717, 7.989504075325736118469432593577
