| L(s) = 1 | + 2·5-s + 5·7-s + 2·11-s + 6·13-s + 3·17-s + 7·19-s + 6·23-s + 3·25-s − 13·29-s − 7·31-s + 10·35-s + 19·37-s + 8·41-s + 2·43-s − 2·47-s + 9·49-s − 7·53-s + 4·55-s + 6·59-s − 21·61-s + 12·65-s + 5·71-s − 6·73-s + 10·77-s + 2·79-s + 12·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.88·7-s + 0.603·11-s + 1.66·13-s + 0.727·17-s + 1.60·19-s + 1.25·23-s + 3/5·25-s − 2.41·29-s − 1.25·31-s + 1.69·35-s + 3.12·37-s + 1.24·41-s + 0.304·43-s − 0.291·47-s + 9/7·49-s − 0.961·53-s + 0.539·55-s + 0.781·59-s − 2.68·61-s + 1.48·65-s + 0.593·71-s − 0.702·73-s + 1.13·77-s + 0.225·79-s + 1.31·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.256999614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.256999614\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 19 T + 160 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 80 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 21 T + 228 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.549764596071347884164245178450, −8.317083417966088551842787040030, −7.77518884452663127012928617492, −7.62552759773609875676005825338, −7.37225970981943262944471597515, −6.86602568759394421452680807508, −6.27777608560631956516505333040, −5.90141708557723876941097491007, −5.60932882720283261153447052646, −5.53876293198186659190058568918, −4.73962338589617553200782360279, −4.72311044467438401711539546030, −3.98934958251338811463628113308, −3.76093671397636364883489823639, −3.06504237899985161399957629674, −2.86281881307806943522262508550, −1.82931664528933569105866262624, −1.81394371783562424830942676410, −1.09133119145956640782465470314, −0.983009697956838234627739818625,
0.983009697956838234627739818625, 1.09133119145956640782465470314, 1.81394371783562424830942676410, 1.82931664528933569105866262624, 2.86281881307806943522262508550, 3.06504237899985161399957629674, 3.76093671397636364883489823639, 3.98934958251338811463628113308, 4.72311044467438401711539546030, 4.73962338589617553200782360279, 5.53876293198186659190058568918, 5.60932882720283261153447052646, 5.90141708557723876941097491007, 6.27777608560631956516505333040, 6.86602568759394421452680807508, 7.37225970981943262944471597515, 7.62552759773609875676005825338, 7.77518884452663127012928617492, 8.317083417966088551842787040030, 8.549764596071347884164245178450
