| L(s) = 1 | + 4-s − 2·9-s − 6·11-s − 2·19-s + 24·29-s + 8·31-s − 2·36-s + 12·41-s − 6·44-s − 2·49-s + 8·61-s − 64-s + 48·71-s − 2·76-s − 20·79-s + 9·81-s + 12·89-s + 12·99-s + 24·101-s − 8·109-s + 24·116-s + 31·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2/3·9-s − 1.80·11-s − 0.458·19-s + 4.45·29-s + 1.43·31-s − 1/3·36-s + 1.87·41-s − 0.904·44-s − 2/7·49-s + 1.02·61-s − 1/8·64-s + 5.69·71-s − 0.229·76-s − 2.25·79-s + 81-s + 1.27·89-s + 1.20·99-s + 2.38·101-s − 0.766·109-s + 2.22·116-s + 2.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213985534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213985534\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 97 T^{2} + 6600 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.228185422941318652328387209003, −8.153479303787426702305880534824, −8.011349811083964745486437480054, −7.79557382174019098625086790552, −7.16317915817777626031155599038, −7.11698354667904398782040743911, −6.81744394986095432480260426126, −6.57523811281186947766622127201, −6.17272196958069003853658844294, −6.13017384145750870749399332576, −5.93435305070075924513073593131, −5.36886183371536140708085593079, −5.15131083899429726382587808062, −4.89865971017124170323198390837, −4.75297236057993523446704824083, −4.29458182247929823610823550982, −4.19430697596854804586292255840, −3.37764189318588346956135593645, −3.36650412365619273646617306105, −2.87595890710021738667560648611, −2.52675734658597325433720398873, −2.33743370097526161974683920623, −2.15328332404131565904666584268, −0.961933834728966628456352324562, −0.789436370322648087206334326454,
0.789436370322648087206334326454, 0.961933834728966628456352324562, 2.15328332404131565904666584268, 2.33743370097526161974683920623, 2.52675734658597325433720398873, 2.87595890710021738667560648611, 3.36650412365619273646617306105, 3.37764189318588346956135593645, 4.19430697596854804586292255840, 4.29458182247929823610823550982, 4.75297236057993523446704824083, 4.89865971017124170323198390837, 5.15131083899429726382587808062, 5.36886183371536140708085593079, 5.93435305070075924513073593131, 6.13017384145750870749399332576, 6.17272196958069003853658844294, 6.57523811281186947766622127201, 6.81744394986095432480260426126, 7.11698354667904398782040743911, 7.16317915817777626031155599038, 7.79557382174019098625086790552, 8.011349811083964745486437480054, 8.153479303787426702305880534824, 8.228185422941318652328387209003
