| L(s) = 1 | − 2·2-s + 2·4-s + 4·5-s − 8·10-s + 6·13-s − 4·16-s + 10·17-s + 8·20-s + 11·25-s − 12·26-s + 8·32-s − 20·34-s − 14·37-s − 22·50-s + 12·52-s + 10·53-s − 24·61-s − 8·64-s + 24·65-s + 20·68-s − 22·73-s + 28·74-s − 16·80-s − 9·81-s + 40·85-s − 20·89-s − 26·97-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.78·5-s − 2.52·10-s + 1.66·13-s − 16-s + 2.42·17-s + 1.78·20-s + 11/5·25-s − 2.35·26-s + 1.41·32-s − 3.42·34-s − 2.30·37-s − 3.11·50-s + 1.66·52-s + 1.37·53-s − 3.07·61-s − 64-s + 2.97·65-s + 2.42·68-s − 2.57·73-s + 3.25·74-s − 1.78·80-s − 81-s + 4.33·85-s − 2.11·89-s − 2.63·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140552436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140552436\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−12.03544996128417207156173930919, −11.74738049845887365154618954305, −10.82796214410643999612947613927, −10.63023964168841980145049918380, −10.19697333540397973062953186803, −9.941594532608218145202032347927, −9.187547350294419413295940592773, −9.131908422051712165194823091286, −8.323425093951738756602128044480, −8.229911823665325104164598562011, −7.20957539847906622450927133130, −7.04059020109515581182621511409, −6.15673848355926331452691721590, −5.73491143410523833211528396224, −5.43462547299117521541803966787, −4.47535795302851131679368478569, −3.45356325176892242942731559183, −2.80530829135999873954974062025, −1.45691168177840979857636693211, −1.45561063744892437789702457079,
1.45561063744892437789702457079, 1.45691168177840979857636693211, 2.80530829135999873954974062025, 3.45356325176892242942731559183, 4.47535795302851131679368478569, 5.43462547299117521541803966787, 5.73491143410523833211528396224, 6.15673848355926331452691721590, 7.04059020109515581182621511409, 7.20957539847906622450927133130, 8.229911823665325104164598562011, 8.323425093951738756602128044480, 9.131908422051712165194823091286, 9.187547350294419413295940592773, 9.941594532608218145202032347927, 10.19697333540397973062953186803, 10.63023964168841980145049918380, 10.82796214410643999612947613927, 11.74738049845887365154618954305, 12.03544996128417207156173930919
