| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 2·9-s + 2·14-s + 16-s − 2·18-s + 25-s + 2·28-s + 4·31-s + 32-s − 2·36-s + 12·41-s + 3·49-s + 50-s + 2·56-s + 4·62-s − 4·63-s + 64-s + 12·71-s − 2·72-s + 16·73-s + 16·79-s − 5·81-s + 12·82-s − 8·97-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.534·14-s + 1/4·16-s − 0.471·18-s + 1/5·25-s + 0.377·28-s + 0.718·31-s + 0.176·32-s − 1/3·36-s + 1.87·41-s + 3/7·49-s + 0.141·50-s + 0.267·56-s + 0.508·62-s − 0.503·63-s + 1/8·64-s + 1.42·71-s − 0.235·72-s + 1.87·73-s + 1.80·79-s − 5/9·81-s + 1.32·82-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.724042514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.724042514\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.301328567380256556725419134120, −8.712752385815154405853330320688, −8.111216784637566461161036134379, −7.917181740448841728738593987127, −7.31586248714206635529021004746, −6.66603545970843839340403555065, −6.26674226361374101090095832265, −5.66536461020403131826870143287, −5.18175866001387178991920947722, −4.72518607098836363650901898297, −4.07342633727944635081172424058, −3.52409018332203014053429639164, −2.66703205583257334869823729676, −2.20688269319052214772231054069, −1.05481391212318198201999949221,
1.05481391212318198201999949221, 2.20688269319052214772231054069, 2.66703205583257334869823729676, 3.52409018332203014053429639164, 4.07342633727944635081172424058, 4.72518607098836363650901898297, 5.18175866001387178991920947722, 5.66536461020403131826870143287, 6.26674226361374101090095832265, 6.66603545970843839340403555065, 7.31586248714206635529021004746, 7.917181740448841728738593987127, 8.111216784637566461161036134379, 8.712752385815154405853330320688, 9.301328567380256556725419134120
