| L(s) = 1 | − 2·9-s + 8·19-s + 4·29-s + 8·31-s + 4·41-s + 6·49-s + 8·59-s − 4·61-s − 8·71-s + 16·79-s − 5·81-s − 20·89-s − 20·101-s + 4·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 16·171-s + 173-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 1.83·19-s + 0.742·29-s + 1.43·31-s + 0.624·41-s + 6/7·49-s + 1.04·59-s − 0.512·61-s − 0.949·71-s + 1.80·79-s − 5/9·81-s − 2.11·89-s − 1.99·101-s + 0.383·109-s − 0.545·121-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 − 0.769·169-s − 1.22·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.296909401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296909401\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 162 T^{2} + p^{2} T^{4} \) |
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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
−10.06977626082370538471893428863, −9.898846866295529902863496427757, −9.251284189612968319728452392367, −8.702394021767180454782889130765, −8.220034691172912677696013631657, −7.67357568433452729615228524518, −7.12428399488025091432258795785, −6.51661267893497595570180523091, −5.84192025295842762081743703128, −5.37319968153162076020779348659, −4.71587371899737625900471590611, −3.95724007252243799902883907405, −3.08173126317146188392605748934, −2.56129487403855568897075165836, −1.11272791949399713419515070834,
1.11272791949399713419515070834, 2.56129487403855568897075165836, 3.08173126317146188392605748934, 3.95724007252243799902883907405, 4.71587371899737625900471590611, 5.37319968153162076020779348659, 5.84192025295842762081743703128, 6.51661267893497595570180523091, 7.12428399488025091432258795785, 7.67357568433452729615228524518, 8.220034691172912677696013631657, 8.702394021767180454782889130765, 9.251284189612968319728452392367, 9.898846866295529902863496427757, 10.06977626082370538471893428863
