| L(s) = 1 | + 2-s + 4-s + 8-s − 6·11-s + 4·13-s + 16-s − 6·22-s − 12·23-s − 10·25-s + 4·26-s + 32-s − 8·37-s − 6·44-s − 12·46-s − 12·47-s − 10·49-s − 10·50-s + 4·52-s + 6·59-s + 16·61-s + 64-s − 24·71-s + 22·73-s − 8·74-s + 24·83-s − 6·88-s − 12·92-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s + 1.10·13-s + 1/4·16-s − 1.27·22-s − 2.50·23-s − 2·25-s + 0.784·26-s + 0.176·32-s − 1.31·37-s − 0.904·44-s − 1.76·46-s − 1.75·47-s − 1.42·49-s − 1.41·50-s + 0.554·52-s + 0.781·59-s + 2.04·61-s + 1/8·64-s − 2.84·71-s + 2.57·73-s − 0.929·74-s + 2.63·83-s − 0.639·88-s − 1.25·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−8.543319215757720981308384286559, −8.255589340941165804909076040404, −7.79453288119723697641112445352, −7.63671497466111343831458484773, −6.69953527334383655382904548898, −6.25936593777468405983226275094, −5.91234365712937538559209143490, −5.13805326110939411578475614907, −5.13777464868847606985261506723, −3.94847330019587176084970982031, −3.86098888246252906299834258338, −3.10746305536689930220572132603, −2.19879385611801234222671009191, −1.80773362483917328069437252787, 0,
1.80773362483917328069437252787, 2.19879385611801234222671009191, 3.10746305536689930220572132603, 3.86098888246252906299834258338, 3.94847330019587176084970982031, 5.13777464868847606985261506723, 5.13805326110939411578475614907, 5.91234365712937538559209143490, 6.25936593777468405983226275094, 6.69953527334383655382904548898, 7.63671497466111343831458484773, 7.79453288119723697641112445352, 8.255589340941165804909076040404, 8.543319215757720981308384286559
