| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 2·9-s − 2·11-s + 4·13-s + 4·15-s + 2·17-s − 4·21-s − 10·23-s + 2·25-s + 6·27-s − 6·29-s − 6·31-s − 4·33-s − 4·35-s + 2·37-s + 8·39-s − 14·41-s + 4·45-s + 2·49-s + 4·51-s − 4·55-s + 10·61-s − 4·63-s + 8·65-s + 24·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 2/3·9-s − 0.603·11-s + 1.10·13-s + 1.03·15-s + 0.485·17-s − 0.872·21-s − 2.08·23-s + 2/5·25-s + 1.15·27-s − 1.11·29-s − 1.07·31-s − 0.696·33-s − 0.676·35-s + 0.328·37-s + 1.28·39-s − 2.18·41-s + 0.596·45-s + 2/7·49-s + 0.560·51-s − 0.539·55-s + 1.28·61-s − 0.503·63-s + 0.992·65-s + 2.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693684995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693684995\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−13.57162418945354658447240790666, −13.12933290606616528754137851184, −12.51324531377935054514399650916, −12.32138059191894184951458441355, −11.22885367011858156334771716748, −10.94847219752289816238378446520, −10.02309333859831312135853099222, −9.878645229577452488950612661783, −9.463644989670771815975227683399, −8.697082895518350614285870132560, −8.188785056306586810555443673500, −7.957488517331540430191804289074, −6.77767091594969739280626834431, −6.64008321229335236841764521920, −5.57372150522651106357303399456, −5.37251201401205682804507580062, −3.89559203824327489931603815983, −3.60686433888271190065560754120, −2.60296196510728259589461946315, −1.82181675459807746972455104933,
1.82181675459807746972455104933, 2.60296196510728259589461946315, 3.60686433888271190065560754120, 3.89559203824327489931603815983, 5.37251201401205682804507580062, 5.57372150522651106357303399456, 6.64008321229335236841764521920, 6.77767091594969739280626834431, 7.957488517331540430191804289074, 8.188785056306586810555443673500, 8.697082895518350614285870132560, 9.463644989670771815975227683399, 9.878645229577452488950612661783, 10.02309333859831312135853099222, 10.94847219752289816238378446520, 11.22885367011858156334771716748, 12.32138059191894184951458441355, 12.51324531377935054514399650916, 13.12933290606616528754137851184, 13.57162418945354658447240790666
