| L(s) = 1 | − 4·11-s + 12·19-s − 10·25-s + 12·41-s + 12·43-s − 14·49-s + 20·59-s + 12·67-s − 4·83-s + 28·107-s + 36·113-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 2.75·19-s − 2·25-s + 1.87·41-s + 1.82·43-s − 2·49-s + 2.60·59-s + 1.46·67-s − 0.439·83-s + 2.70·107-s + 3.38·113-s + 8/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.991058604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.991058604\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.244811999508504807312081426048, −8.120716373859945936782177859829, −7.68220535758383737115108735238, −7.58552329396578604947500324913, −7.02022457367432300363253741052, −6.98290475216206760724877591999, −6.00174647285035267565268072896, −5.99442003471944340044173645962, −5.52434031702028917311817113157, −5.37304250980212113549721952114, −4.71566593156946449736971326421, −4.57264196491881478097447338302, −3.75717623548078198902051292955, −3.68997200021040974890985276774, −3.04003305886084407618584265366, −2.75524989010618103348991005732, −2.13657106967942128584931039402, −1.82743599049118109263536201005, −0.854154471349132173763761623897, −0.62870428609346320051693236345,
0.62870428609346320051693236345, 0.854154471349132173763761623897, 1.82743599049118109263536201005, 2.13657106967942128584931039402, 2.75524989010618103348991005732, 3.04003305886084407618584265366, 3.68997200021040974890985276774, 3.75717623548078198902051292955, 4.57264196491881478097447338302, 4.71566593156946449736971326421, 5.37304250980212113549721952114, 5.52434031702028917311817113157, 5.99442003471944340044173645962, 6.00174647285035267565268072896, 6.98290475216206760724877591999, 7.02022457367432300363253741052, 7.58552329396578604947500324913, 7.68220535758383737115108735238, 8.120716373859945936782177859829, 8.244811999508504807312081426048
