| L(s) = 1 | + 3-s − 8·5-s + 2·7-s + 4·11-s − 8·15-s − 2·17-s + 2·19-s + 2·21-s + 38·25-s − 27-s + 6·29-s − 20·31-s + 4·33-s − 16·35-s − 10·37-s − 8·41-s − 4·43-s − 8·47-s + 7·49-s − 2·51-s − 20·53-s − 32·55-s + 2·57-s + 8·59-s + 14·61-s − 2·67-s − 16·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3.57·5-s + 0.755·7-s + 1.20·11-s − 2.06·15-s − 0.485·17-s + 0.458·19-s + 0.436·21-s + 38/5·25-s − 0.192·27-s + 1.11·29-s − 3.59·31-s + 0.696·33-s − 2.70·35-s − 1.64·37-s − 1.24·41-s − 0.609·43-s − 1.16·47-s + 49-s − 0.280·51-s − 2.74·53-s − 4.31·55-s + 0.264·57-s + 1.04·59-s + 1.79·61-s − 0.244·67-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 16 T + 185 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.726624634601880262540632241067, −8.539825990130208982825812864153, −8.062270280186357456188813329597, −8.020775433687687247272169423144, −7.27112693309616845244501986591, −7.15740809612279037781928137880, −6.98904904998467218003204509059, −6.50205386759873628864082251007, −5.48163926250373213158095886576, −5.29642770170814678532703623883, −4.46309268594975672450750702001, −4.46014437005376529873956901530, −3.94470945639180785966003757093, −3.57600215762941453280140440722, −3.23790575075785422133145951312, −2.90517411398229217648747621974, −1.66193045617956345981245735836, −1.38837792382931733151992885352, 0, 0,
1.38837792382931733151992885352, 1.66193045617956345981245735836, 2.90517411398229217648747621974, 3.23790575075785422133145951312, 3.57600215762941453280140440722, 3.94470945639180785966003757093, 4.46014437005376529873956901530, 4.46309268594975672450750702001, 5.29642770170814678532703623883, 5.48163926250373213158095886576, 6.50205386759873628864082251007, 6.98904904998467218003204509059, 7.15740809612279037781928137880, 7.27112693309616845244501986591, 8.020775433687687247272169423144, 8.062270280186357456188813329597, 8.539825990130208982825812864153, 8.726624634601880262540632241067
