| L(s) = 1 | + 4-s − 8·7-s − 8·13-s + 16-s + 10·19-s + 25-s − 8·28-s + 4·31-s − 8·37-s + 22·43-s + 34·49-s − 8·52-s − 20·61-s + 64-s + 10·67-s − 14·73-s + 10·76-s + 28·79-s + 64·91-s + 22·97-s + 100-s − 8·103-s − 8·109-s − 8·112-s − 13·121-s + 4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.02·7-s − 2.21·13-s + 1/4·16-s + 2.29·19-s + 1/5·25-s − 1.51·28-s + 0.718·31-s − 1.31·37-s + 3.35·43-s + 34/7·49-s − 1.10·52-s − 2.56·61-s + 1/8·64-s + 1.22·67-s − 1.63·73-s + 1.14·76-s + 3.15·79-s + 6.70·91-s + 2.23·97-s + 1/10·100-s − 0.788·103-s − 0.766·109-s − 0.755·112-s − 1.18·121-s + 0.359·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77404691036140567875834676085, −7.55386549137356459500397001290, −7.23347773420758791578828913025, −6.86193526564664030360784937729, −6.22301446970872085611681593876, −6.12943902760099620494315213595, −5.38292631066779946248479008415, −5.04061555557317003360241923294, −4.29111105534765483466139869890, −3.47204628573842727701429029278, −3.28121157426783676388528803494, −2.64427174967641130853789427610, −2.41912201488827531235600451311, −0.949027824987604272203929795769, 0,
0.949027824987604272203929795769, 2.41912201488827531235600451311, 2.64427174967641130853789427610, 3.28121157426783676388528803494, 3.47204628573842727701429029278, 4.29111105534765483466139869890, 5.04061555557317003360241923294, 5.38292631066779946248479008415, 6.12943902760099620494315213595, 6.22301446970872085611681593876, 6.86193526564664030360784937729, 7.23347773420758791578828913025, 7.55386549137356459500397001290, 7.77404691036140567875834676085
