| L(s) = 1 | − 3·3-s − 7-s + 6·9-s − 3·11-s − 3·17-s − 8·19-s + 3·21-s − 5·25-s − 9·27-s − 6·29-s − 15·31-s + 9·33-s − 6·41-s − 2·43-s − 6·47-s + 7·49-s + 9·51-s − 9·53-s + 24·57-s − 12·59-s + 2·61-s − 6·63-s − 15·71-s − 11·73-s + 15·75-s + 3·77-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s − 0.904·11-s − 0.727·17-s − 1.83·19-s + 0.654·21-s − 25-s − 1.73·27-s − 1.11·29-s − 2.69·31-s + 1.56·33-s − 0.937·41-s − 0.304·43-s − 0.875·47-s + 49-s + 1.26·51-s − 1.23·53-s + 3.17·57-s − 1.56·59-s + 0.256·61-s − 0.755·63-s − 1.78·71-s − 1.28·73-s + 1.73·75-s + 0.341·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
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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
−10.30281629806444633501313277491, −10.19258121219974308753277069009, −9.362761273008386626364611519566, −9.166452827706020649598959887187, −8.590694427399453822978946181539, −7.964139812415366027886769100609, −7.50421612284770897068138909115, −7.06460943524708871249096481191, −6.68316093453858192368193457428, −6.08487823422186698921886093926, −5.77375839691224597568141315212, −5.48574022548231221319455412900, −4.63795670608312185256000565804, −4.55078163082162451713484139177, −3.77273247850377530370214520616, −3.21313761629033417116307558991, −2.02472160294981593956991087113, −1.78273877438761374920081023526, 0, 0,
1.78273877438761374920081023526, 2.02472160294981593956991087113, 3.21313761629033417116307558991, 3.77273247850377530370214520616, 4.55078163082162451713484139177, 4.63795670608312185256000565804, 5.48574022548231221319455412900, 5.77375839691224597568141315212, 6.08487823422186698921886093926, 6.68316093453858192368193457428, 7.06460943524708871249096481191, 7.50421612284770897068138909115, 7.964139812415366027886769100609, 8.590694427399453822978946181539, 9.166452827706020649598959887187, 9.362761273008386626364611519566, 10.19258121219974308753277069009, 10.30281629806444633501313277491
