L(s) = 1 | + 4-s + 2·5-s + 9-s − 11-s − 3·16-s + 2·20-s + 8·23-s + 3·25-s − 16·31-s + 36-s + 12·37-s − 44-s + 2·45-s + 8·47-s + 2·49-s − 4·53-s − 2·55-s + 8·59-s − 7·64-s + 16·67-s − 6·80-s + 81-s + 4·89-s + 8·92-s + 4·97-s − 99-s + 3·100-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 3/4·16-s + 0.447·20-s + 1.66·23-s + 3/5·25-s − 2.87·31-s + 1/6·36-s + 1.97·37-s − 0.150·44-s + 0.298·45-s + 1.16·47-s + 2/7·49-s − 0.549·53-s − 0.269·55-s + 1.04·59-s − 7/8·64-s + 1.95·67-s − 0.670·80-s + 1/9·81-s + 0.423·89-s + 0.834·92-s + 0.406·97-s − 0.100·99-s + 3/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.447461377\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.447461377\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.946456430300616018933661220162, −8.498244207061406232686315811105, −7.72297200030437951699351708002, −7.33142122310648485270568820369, −7.01648641519502427315426533965, −6.50145107649116507165939720580, −5.96394982864770564128835187499, −5.48436797339662241268545514269, −5.04535994307406662757349957039, −4.46241986760257866122396361641, −3.75668104286618219930554973290, −3.10184840890448583577983619095, −2.33940350691560819237429097311, −1.99466443651709397359232876719, −0.940395419944072625461824116915,
0.940395419944072625461824116915, 1.99466443651709397359232876719, 2.33940350691560819237429097311, 3.10184840890448583577983619095, 3.75668104286618219930554973290, 4.46241986760257866122396361641, 5.04535994307406662757349957039, 5.48436797339662241268545514269, 5.96394982864770564128835187499, 6.50145107649116507165939720580, 7.01648641519502427315426533965, 7.33142122310648485270568820369, 7.72297200030437951699351708002, 8.498244207061406232686315811105, 8.946456430300616018933661220162