| L(s) = 1 | − 3-s + 9-s − 4·11-s + 2·17-s − 8·19-s + 6·25-s − 27-s + 4·33-s − 6·41-s − 12·43-s + 2·49-s − 2·51-s + 8·57-s + 4·59-s + 8·67-s + 4·73-s − 6·75-s + 81-s + 12·83-s + 2·89-s − 16·97-s − 4·99-s + 20·107-s + 10·113-s − 10·121-s + 6·123-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.485·17-s − 1.83·19-s + 6/5·25-s − 0.192·27-s + 0.696·33-s − 0.937·41-s − 1.82·43-s + 2/7·49-s − 0.280·51-s + 1.05·57-s + 0.520·59-s + 0.977·67-s + 0.468·73-s − 0.692·75-s + 1/9·81-s + 1.31·83-s + 0.211·89-s − 1.62·97-s − 0.402·99-s + 1.93·107-s + 0.940·113-s − 0.909·121-s + 0.541·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9790939420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9790939420\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.504260956025760287998725072767, −8.174291916014102954935419558366, −7.76139333301962742643024545645, −7.08634357061706526731867665475, −6.66435669311267245708484217317, −6.43546144631690904479700213719, −5.64636536881997697230542212926, −5.34318718203313404680289768503, −4.79196669428260128340004223891, −4.43417520195675120973707277560, −3.64043174116040341978914976440, −3.11035369176159785408075083066, −2.35010546621296096213549478531, −1.74602073288839591527261259305, −0.54245484583042170863001187940,
0.54245484583042170863001187940, 1.74602073288839591527261259305, 2.35010546621296096213549478531, 3.11035369176159785408075083066, 3.64043174116040341978914976440, 4.43417520195675120973707277560, 4.79196669428260128340004223891, 5.34318718203313404680289768503, 5.64636536881997697230542212926, 6.43546144631690904479700213719, 6.66435669311267245708484217317, 7.08634357061706526731867665475, 7.76139333301962742643024545645, 8.174291916014102954935419558366, 8.504260956025760287998725072767
