| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 9-s − 2·10-s − 2·13-s + 16-s − 6·17-s + 18-s + 2·20-s + 3·25-s + 2·26-s − 2·29-s − 32-s + 6·34-s − 36-s + 8·37-s − 2·40-s − 4·41-s − 2·45-s − 6·49-s − 3·50-s − 2·52-s − 8·53-s + 2·58-s + 20·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 1/3·9-s − 0.632·10-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.447·20-s + 3/5·25-s + 0.392·26-s − 0.371·29-s − 0.176·32-s + 1.02·34-s − 1/6·36-s + 1.31·37-s − 0.316·40-s − 0.624·41-s − 0.298·45-s − 6/7·49-s − 0.424·50-s − 0.277·52-s − 1.09·53-s + 0.262·58-s + 2.56·61-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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.908766527285789705494532922641, −7.36994465965842122346967934906, −6.82384735977667288550221176370, −6.59846869601971368962532383618, −6.20017975584315866536072333167, −5.61524320910454177391970513410, −5.22754972365114841635336652901, −4.71965166171276365940589823165, −4.18541982786241713118589886609, −3.51257151715482421324557938471, −2.84469967484086300371970894042, −2.29930709472591036091209413783, −1.97717753435002883088513428658, −1.07369292935199752555088435464, 0,
1.07369292935199752555088435464, 1.97717753435002883088513428658, 2.29930709472591036091209413783, 2.84469967484086300371970894042, 3.51257151715482421324557938471, 4.18541982786241713118589886609, 4.71965166171276365940589823165, 5.22754972365114841635336652901, 5.61524320910454177391970513410, 6.20017975584315866536072333167, 6.59846869601971368962532383618, 6.82384735977667288550221176370, 7.36994465965842122346967934906, 7.908766527285789705494532922641
