| L(s) = 1 | + 4-s − 9-s + 8·11-s + 2·19-s − 36-s − 4·41-s + 8·44-s − 4·49-s − 2·59-s − 64-s + 2·76-s + 81-s − 2·89-s − 8·99-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 2·171-s + ⋯ |
| L(s) = 1 | + 4-s − 9-s + 8·11-s + 2·19-s − 36-s − 4·41-s + 8·44-s − 4·49-s − 2·59-s − 64-s + 2·76-s + 81-s − 2·89-s − 8·99-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 2·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.001015271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.001015271\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_1$ | \( ( 1 - T )^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.44928194747467737429570350080, −5.98390025862653471452153568165, −5.94211625349037083324616459568, −5.79209547458564330981075683371, −5.46632152122279218893161858828, −5.38609833290775763687345843884, −4.77457154209691307090296605882, −4.72572875802481782931787459111, −4.52170086489716730676041622548, −4.35857555064817250045099685543, −4.30191325993526071717233003923, −3.75402158975689764662214252648, −3.68820096148790352302378910949, −3.46295865095758845293573209343, −3.43768681984658665990776439560, −3.18225631621518551137418978612, −2.99780975261159039440143665940, −2.91197089026125269057044451346, −1.97544762907187067970663991090, −1.91145946980972668599271943336, −1.68814860178769907810409759187, −1.55370972081223265993847189423, −1.51420149393272148958078056702, −0.973418603987925881074629323052, −0.914124252531386070341280746460,
0.914124252531386070341280746460, 0.973418603987925881074629323052, 1.51420149393272148958078056702, 1.55370972081223265993847189423, 1.68814860178769907810409759187, 1.91145946980972668599271943336, 1.97544762907187067970663991090, 2.91197089026125269057044451346, 2.99780975261159039440143665940, 3.18225631621518551137418978612, 3.43768681984658665990776439560, 3.46295865095758845293573209343, 3.68820096148790352302378910949, 3.75402158975689764662214252648, 4.30191325993526071717233003923, 4.35857555064817250045099685543, 4.52170086489716730676041622548, 4.72572875802481782931787459111, 4.77457154209691307090296605882, 5.38609833290775763687345843884, 5.46632152122279218893161858828, 5.79209547458564330981075683371, 5.94211625349037083324616459568, 5.98390025862653471452153568165, 6.44928194747467737429570350080
