| L(s) = 1 | − 2·3-s − 2·5-s − 9-s − 2·11-s − 2·13-s + 4·15-s − 4·17-s − 12·19-s + 4·23-s − 7·25-s + 6·27-s + 2·29-s − 6·31-s + 4·33-s − 8·37-s + 4·39-s + 8·41-s − 10·43-s + 2·45-s − 2·47-s − 6·49-s + 8·51-s + 2·53-s + 4·55-s + 24·57-s − 4·59-s − 4·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.03·15-s − 0.970·17-s − 2.75·19-s + 0.834·23-s − 7/5·25-s + 1.15·27-s + 0.371·29-s − 1.07·31-s + 0.696·33-s − 1.31·37-s + 0.640·39-s + 1.24·41-s − 1.52·43-s + 0.298·45-s − 0.291·47-s − 6/7·49-s + 1.12·51-s + 0.274·53-s + 0.539·55-s + 3.17·57-s − 0.520·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 21 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 77 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 157 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.79994269815468509870656367353, −10.77940257591454959174509561423, −10.04635000379999031858422786551, −9.583764713617372180064981145552, −8.748088184287142570196603978147, −8.694505958221330306125194405903, −8.039650158973791495849288991815, −7.72629457534308639666661223025, −6.83453423296081787662265008717, −6.77113447264978577334693149419, −6.05295460422387584796484389668, −5.71336715935644774872591715769, −4.94519557362189222641599211370, −4.71844359536839924702887466152, −4.01573819238036982881848647110, −3.47137365315722201634130699142, −2.50531499972131718504053456344, −1.93472928332123355893500125396, 0, 0,
1.93472928332123355893500125396, 2.50531499972131718504053456344, 3.47137365315722201634130699142, 4.01573819238036982881848647110, 4.71844359536839924702887466152, 4.94519557362189222641599211370, 5.71336715935644774872591715769, 6.05295460422387584796484389668, 6.77113447264978577334693149419, 6.83453423296081787662265008717, 7.72629457534308639666661223025, 8.039650158973791495849288991815, 8.694505958221330306125194405903, 8.748088184287142570196603978147, 9.583764713617372180064981145552, 10.04635000379999031858422786551, 10.77940257591454959174509561423, 10.79994269815468509870656367353
