| L(s) = 1 | + 4·4-s + 4·7-s − 6·9-s + 12·16-s − 4·25-s + 16·28-s − 24·36-s − 2·49-s − 24·63-s + 32·64-s + 40·79-s + 27·81-s − 16·100-s + 48·112-s + 20·121-s + 127-s + 131-s + 137-s + 139-s − 72·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.51·7-s − 2·9-s + 3·16-s − 4/5·25-s + 3.02·28-s − 4·36-s − 2/7·49-s − 3.02·63-s + 4·64-s + 4.50·79-s + 3·81-s − 8/5·100-s + 4.53·112-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.370212305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.370212305\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.463905962790474327481766856271, −8.918599271963746007343294473817, −8.731907194950073710484927293545, −8.697504579416260637578776703231, −7.991243535936139077169952163440, −7.915212305677035958632268223823, −7.88548266371998143322691787302, −7.62790156843130771759978278948, −7.32251673755100357297580495237, −6.62858408236905048945541561093, −6.48828852458800911734825677052, −6.44847214491548506667101388735, −6.08815895988233603287945428027, −5.61829327842256544630642368453, −5.26155228062265800101000909279, −5.23929776029237819618289289337, −4.93537919520615112704814041837, −4.23884153285095206597732003327, −3.81375988599753106073857001856, −3.39483135222866222223506198993, −3.12584435405125948189141999492, −2.51057144729191734177692515939, −2.30097137970135329042072321867, −1.88609458690155200969310678018, −1.19755069238335430803530607186,
1.19755069238335430803530607186, 1.88609458690155200969310678018, 2.30097137970135329042072321867, 2.51057144729191734177692515939, 3.12584435405125948189141999492, 3.39483135222866222223506198993, 3.81375988599753106073857001856, 4.23884153285095206597732003327, 4.93537919520615112704814041837, 5.23929776029237819618289289337, 5.26155228062265800101000909279, 5.61829327842256544630642368453, 6.08815895988233603287945428027, 6.44847214491548506667101388735, 6.48828852458800911734825677052, 6.62858408236905048945541561093, 7.32251673755100357297580495237, 7.62790156843130771759978278948, 7.88548266371998143322691787302, 7.915212305677035958632268223823, 7.991243535936139077169952163440, 8.697504579416260637578776703231, 8.731907194950073710484927293545, 8.918599271963746007343294473817, 9.463905962790474327481766856271
