| L(s) = 1 | − 2·5-s + 4·13-s − 25-s − 16·31-s + 20·37-s + 12·41-s − 8·43-s + 10·49-s − 20·53-s − 8·65-s + 24·67-s − 32·79-s + 8·83-s − 20·89-s − 40·107-s + 18·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.10·13-s − 1/5·25-s − 2.87·31-s + 3.28·37-s + 1.87·41-s − 1.21·43-s + 10/7·49-s − 2.74·53-s − 0.992·65-s + 2.93·67-s − 3.60·79-s + 0.878·83-s − 2.11·89-s − 3.86·107-s + 1.63·121-s + 1.07·125-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 + 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.368713514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368713514\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.162995944366059425466767799339, −8.387134035990489276766665418250, −8.190365744772538123403888906739, −7.910449631036686579642143372138, −7.52000547906358510361120980502, −7.05396303804935824699683401699, −6.88922716980739344009202808527, −6.06951324130777776679767519808, −6.01871320025884937751089199515, −5.61777388354952501180826228999, −5.10412056097083326860946758494, −4.54022443721985578394127477529, −4.08224064862707085530631010825, −3.90153461380821647862837790970, −3.50805525049656096733039729377, −2.81298082593325811837869649026, −2.51531953082232572790009597797, −1.68190882258874019825918592571, −1.22843582981237085624879019230, −0.40136974287515611266036619101,
0.40136974287515611266036619101, 1.22843582981237085624879019230, 1.68190882258874019825918592571, 2.51531953082232572790009597797, 2.81298082593325811837869649026, 3.50805525049656096733039729377, 3.90153461380821647862837790970, 4.08224064862707085530631010825, 4.54022443721985578394127477529, 5.10412056097083326860946758494, 5.61777388354952501180826228999, 6.01871320025884937751089199515, 6.06951324130777776679767519808, 6.88922716980739344009202808527, 7.05396303804935824699683401699, 7.52000547906358510361120980502, 7.910449631036686579642143372138, 8.190365744772538123403888906739, 8.387134035990489276766665418250, 9.162995944366059425466767799339
