L(s) = 1 | + 4-s + 13-s + 16-s + 4·19-s + 8·25-s − 8·31-s − 5·37-s + 7·43-s − 6·49-s + 52-s + 7·61-s + 64-s − 17·67-s + 13·73-s + 4·76-s − 11·79-s − 20·97-s + 8·100-s − 8·103-s − 14·109-s − 4·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 0.277·13-s + 1/4·16-s + 0.917·19-s + 8/5·25-s − 1.43·31-s − 0.821·37-s + 1.06·43-s − 6/7·49-s + 0.138·52-s + 0.896·61-s + 1/8·64-s − 2.07·67-s + 1.52·73-s + 0.458·76-s − 1.23·79-s − 2.03·97-s + 4/5·100-s − 0.788·103-s − 1.34·109-s − 0.363·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.410·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20412 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20412 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.321137289\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.321137289\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.78054444854833913410674012940, −10.55376540537036846977385905472, −9.731817561821762417591946518018, −9.258065637262344174182183234024, −8.727186239206226935472447794602, −8.103965047141443897493173295234, −7.43511443565168193777003995180, −6.98507294544779013933608971902, −6.43427702798258515878327163861, −5.58223271934365262380028520310, −5.19338692173146828831048630873, −4.24264059355529231234305244599, −3.40312992704562422441566083600, −2.68585712044940757410764993903, −1.44784256710938688824185178465,
1.44784256710938688824185178465, 2.68585712044940757410764993903, 3.40312992704562422441566083600, 4.24264059355529231234305244599, 5.19338692173146828831048630873, 5.58223271934365262380028520310, 6.43427702798258515878327163861, 6.98507294544779013933608971902, 7.43511443565168193777003995180, 8.103965047141443897493173295234, 8.727186239206226935472447794602, 9.258065637262344174182183234024, 9.731817561821762417591946518018, 10.55376540537036846977385905472, 10.78054444854833913410674012940