| L(s) = 1 | + 2-s + 4-s − 2·5-s − 3·7-s + 8-s − 5·9-s − 2·10-s − 11-s − 3·13-s − 3·14-s + 16-s − 5·18-s − 2·20-s − 22-s − 2·25-s − 3·26-s − 3·28-s + 31-s + 32-s + 6·35-s − 5·36-s − 2·40-s − 2·43-s − 44-s + 10·45-s − 6·47-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.13·7-s + 0.353·8-s − 5/3·9-s − 0.632·10-s − 0.301·11-s − 0.832·13-s − 0.801·14-s + 1/4·16-s − 1.17·18-s − 0.447·20-s − 0.213·22-s − 2/5·25-s − 0.588·26-s − 0.566·28-s + 0.179·31-s + 0.176·32-s + 1.01·35-s − 5/6·36-s − 0.316·40-s − 0.304·43-s − 0.150·44-s + 1.49·45-s − 0.875·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
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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
−10.30625158281814654646136601008, −9.662116873776763503704987997104, −9.290253827775280498435333499559, −8.389606765016871261693289332634, −8.163601736289340551621444198358, −7.49141551790521137416786289005, −6.81577031659550740336496387543, −6.38032383064270632520363812271, −5.60956005351140354594733538219, −5.25056001774719032595567424552, −4.35698877117903131179237458107, −3.60640018363373500720888002371, −3.04296613229166822081277949951, −2.40461908035731444788935104109, 0,
2.40461908035731444788935104109, 3.04296613229166822081277949951, 3.60640018363373500720888002371, 4.35698877117903131179237458107, 5.25056001774719032595567424552, 5.60956005351140354594733538219, 6.38032383064270632520363812271, 6.81577031659550740336496387543, 7.49141551790521137416786289005, 8.163601736289340551621444198358, 8.389606765016871261693289332634, 9.290253827775280498435333499559, 9.662116873776763503704987997104, 10.30625158281814654646136601008
