| L(s) = 1 | − 4·7-s − 9-s + 8·11-s + 4·23-s + 2·29-s + 4·37-s − 12·43-s + 9·49-s + 8·53-s + 4·63-s + 8·67-s + 16·71-s − 32·77-s + 16·79-s − 8·81-s − 8·99-s + 16·107-s + 22·109-s + 4·113-s + 27·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1/3·9-s + 2.41·11-s + 0.834·23-s + 0.371·29-s + 0.657·37-s − 1.82·43-s + 9/7·49-s + 1.09·53-s + 0.503·63-s + 0.977·67-s + 1.89·71-s − 3.64·77-s + 1.80·79-s − 8/9·81-s − 0.804·99-s + 1.54·107-s + 2.10·109-s + 0.376·113-s + 2.45·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.034964418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034964418\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
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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
−7.68313722963231684362247249288, −7.18037030660825335804444992294, −6.76792278080776726528934411600, −6.45245502129997290332629657025, −6.32575800546953806715373001893, −5.77931717819679328828939194457, −5.12228092368792626308831320445, −4.75450575046833732417670585319, −3.98792211302815590430187104704, −3.69206353960463104183727705685, −3.39711237700794286554640589971, −2.75526219452087525683394391129, −2.12067515740541168954849734392, −1.27924561943844186654989996949, −0.64380406187411459918854376673,
0.64380406187411459918854376673, 1.27924561943844186654989996949, 2.12067515740541168954849734392, 2.75526219452087525683394391129, 3.39711237700794286554640589971, 3.69206353960463104183727705685, 3.98792211302815590430187104704, 4.75450575046833732417670585319, 5.12228092368792626308831320445, 5.77931717819679328828939194457, 6.32575800546953806715373001893, 6.45245502129997290332629657025, 6.76792278080776726528934411600, 7.18037030660825335804444992294, 7.68313722963231684362247249288
