| L(s) = 1 | + 2·5-s − 2·9-s + 2·13-s + 4·17-s + 3·25-s − 12·29-s − 12·37-s + 4·41-s − 4·45-s − 14·49-s − 4·53-s − 28·61-s + 4·65-s − 4·73-s − 5·81-s + 8·85-s − 12·89-s + 4·97-s − 20·101-s − 20·109-s + 4·113-s − 4·117-s − 18·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2/3·9-s + 0.554·13-s + 0.970·17-s + 3/5·25-s − 2.22·29-s − 1.97·37-s + 0.624·41-s − 0.596·45-s − 2·49-s − 0.549·53-s − 3.58·61-s + 0.496·65-s − 0.468·73-s − 5/9·81-s + 0.867·85-s − 1.27·89-s + 0.406·97-s − 1.99·101-s − 1.91·109-s + 0.376·113-s − 0.369·117-s − 1.63·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.282388148593036711889740506020, −7.76074024039613588419363915341, −7.44095337164983387150060505996, −6.79685282000922095191914374444, −6.33588991532526860429499225622, −5.89482538800335589915814344399, −5.43018572527764567927669449080, −5.24172374409154567149889570954, −4.42926641381854419615826380073, −3.83609522350488929158971696987, −3.10020205606080864107700760801, −2.93871105322865322021254198939, −1.66550824653592893392121562696, −1.63546887865200897104772810976, 0,
1.63546887865200897104772810976, 1.66550824653592893392121562696, 2.93871105322865322021254198939, 3.10020205606080864107700760801, 3.83609522350488929158971696987, 4.42926641381854419615826380073, 5.24172374409154567149889570954, 5.43018572527764567927669449080, 5.89482538800335589915814344399, 6.33588991532526860429499225622, 6.79685282000922095191914374444, 7.44095337164983387150060505996, 7.76074024039613588419363915341, 8.282388148593036711889740506020
