| L(s) = 1 | − 3·3-s + 6·9-s − 7·19-s − 5·25-s − 9·27-s + 13·49-s + 21·57-s − 21·67-s − 24·73-s + 15·75-s + 9·81-s − 11·121-s + 127-s + 131-s + 137-s + 139-s − 39·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s − 42·171-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 1.60·19-s − 25-s − 1.73·27-s + 13/7·49-s + 2.78·57-s − 2.56·67-s − 2.80·73-s + 1.73·75-s + 81-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.21·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.76·169-s − 3.21·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.03289735748629283952937134769, −9.270981567679906681995908952508, −8.784774390971334014644934694365, −8.189478920900444648462771977182, −7.36990395992700175941088900995, −7.14839517916630981371693058846, −6.37808766560145263834313988983, −5.90629154350294878239687671104, −5.69678869113846324851276172303, −4.74937257504541601053316791273, −4.39738860790570192073165449065, −3.74799988141654562579655820524, −2.51332371484229190745930153240, −1.47089399278593014390941848189, 0,
1.47089399278593014390941848189, 2.51332371484229190745930153240, 3.74799988141654562579655820524, 4.39738860790570192073165449065, 4.74937257504541601053316791273, 5.69678869113846324851276172303, 5.90629154350294878239687671104, 6.37808766560145263834313988983, 7.14839517916630981371693058846, 7.36990395992700175941088900995, 8.189478920900444648462771977182, 8.784774390971334014644934694365, 9.270981567679906681995908952508, 10.03289735748629283952937134769
