| L(s) = 1 | + 2·9-s − 4·13-s + 2·17-s − 8·19-s + 10·25-s + 24·43-s + 16·47-s + 10·49-s + 12·53-s − 8·59-s − 8·67-s − 5·81-s + 8·83-s − 28·89-s − 20·101-s + 16·103-s − 8·117-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 1.10·13-s + 0.485·17-s − 1.83·19-s + 2·25-s + 3.65·43-s + 2.33·47-s + 10/7·49-s + 1.64·53-s − 1.04·59-s − 0.977·67-s − 5/9·81-s + 0.878·83-s − 2.96·89-s − 1.99·101-s + 1.57·103-s − 0.739·117-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455071453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455071453\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−12.45173620367729600128500310070, −11.81979498009659778183415607023, −10.83862963460453809680211455226, −10.79775067712982601333825348438, −10.44403948167659640158134581646, −9.776256155119552880110608634349, −9.289046480681137100625397339415, −8.700721178111605961356860273576, −8.547041723071807033843932199956, −7.48395248870592588027510989082, −7.30723505835557254261518033070, −6.94592676285430572717045162756, −5.99033994894565169407786853753, −5.75785155140974070054336847754, −4.86096007328488008531224740394, −4.30107750003002185772892906206, −3.97507305024025642209311452288, −2.67943085351653375449713383125, −2.40026718912510933445283289055, −0.996228684884488632246509041931,
0.996228684884488632246509041931, 2.40026718912510933445283289055, 2.67943085351653375449713383125, 3.97507305024025642209311452288, 4.30107750003002185772892906206, 4.86096007328488008531224740394, 5.75785155140974070054336847754, 5.99033994894565169407786853753, 6.94592676285430572717045162756, 7.30723505835557254261518033070, 7.48395248870592588027510989082, 8.547041723071807033843932199956, 8.700721178111605961356860273576, 9.289046480681137100625397339415, 9.776256155119552880110608634349, 10.44403948167659640158134581646, 10.79775067712982601333825348438, 10.83862963460453809680211455226, 11.81979498009659778183415607023, 12.45173620367729600128500310070
