L(s) = 1 | + 2·5-s + 4·7-s − 2·11-s + 8·17-s + 8·19-s − 8·23-s + 3·25-s + 4·29-s + 8·35-s + 12·37-s − 4·41-s + 12·43-s − 8·47-s + 6·49-s + 4·53-s − 4·55-s − 8·59-s − 12·61-s + 16·71-s − 8·77-s − 20·83-s + 16·85-s + 4·89-s + 16·95-s + 12·97-s − 4·101-s − 16·103-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.603·11-s + 1.94·17-s + 1.83·19-s − 1.66·23-s + 3/5·25-s + 0.742·29-s + 1.35·35-s + 1.97·37-s − 0.624·41-s + 1.82·43-s − 1.16·47-s + 6/7·49-s + 0.549·53-s − 0.539·55-s − 1.04·59-s − 1.53·61-s + 1.89·71-s − 0.911·77-s − 2.19·83-s + 1.73·85-s + 0.423·89-s + 1.64·95-s + 1.21·97-s − 0.398·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.362078992\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.362078992\) |
\(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.965381801156168283272471023746, −7.64370032542845224832630795707, −7.36149128963944271334102410866, −7.36095641566394068722333768797, −6.34615805811010462160664146287, −6.31981750987947601465700547473, −5.83607366290488814726543059911, −5.55527703368403334649323090761, −5.28952618023231337302848626887, −4.95400421429265281698702619064, −4.53368275081722657082823354252, −4.26923387341740387707431209998, −3.63495828976766875045445310362, −3.28085278881156929025595108900, −2.75780331969216714585310815898, −2.56074077660263243698300210894, −1.84624113797027591371481666839, −1.57212968246214759875529195810, −1.08216848792811576668245872007, −0.65584218080414115825377466433,
0.65584218080414115825377466433, 1.08216848792811576668245872007, 1.57212968246214759875529195810, 1.84624113797027591371481666839, 2.56074077660263243698300210894, 2.75780331969216714585310815898, 3.28085278881156929025595108900, 3.63495828976766875045445310362, 4.26923387341740387707431209998, 4.53368275081722657082823354252, 4.95400421429265281698702619064, 5.28952618023231337302848626887, 5.55527703368403334649323090761, 5.83607366290488814726543059911, 6.31981750987947601465700547473, 6.34615805811010462160664146287, 7.36095641566394068722333768797, 7.36149128963944271334102410866, 7.64370032542845224832630795707, 7.965381801156168283272471023746