| L(s) = 1 | + 4·4-s − 3·9-s + 4·16-s − 12·19-s − 30·31-s − 12·36-s − 2·49-s + 54·61-s − 16·64-s − 48·76-s + 34·79-s − 4·109-s − 22·121-s − 120·124-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 36·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2·4-s − 9-s + 16-s − 2.75·19-s − 5.38·31-s − 2·36-s − 2/7·49-s + 6.91·61-s − 2·64-s − 5.50·76-s + 3.82·79-s − 0.383·109-s − 2·121-s − 10.7·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 2.75·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.200875636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.200875636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 167 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.61490590206857053901811644296, −7.60567950771521789978397705076, −7.38226228244397334552178217688, −6.89258776389101913633102092229, −6.79708566431453852598271358562, −6.75151195981889867314297622062, −6.63255680361938415060013258247, −6.12751598329163930926402689935, −5.83241277213078958917455451227, −5.82439314737636279219987513629, −5.37956984663547349741553726609, −5.26546767610340386199346250864, −4.92203804438125853298384469613, −4.66220408144396368763745782097, −3.94955173303974419353559382745, −3.90193060525851594972408428590, −3.67769329628755774515261565780, −3.54100438124106734535074497064, −2.94234495738880880291176160224, −2.42035122585773161268303658130, −2.22702136829937267538182573511, −2.18997881710794867244056078055, −1.96086430408810252112906155511, −1.33675024771633153070788780216, −0.30359770373932817494901780829,
0.30359770373932817494901780829, 1.33675024771633153070788780216, 1.96086430408810252112906155511, 2.18997881710794867244056078055, 2.22702136829937267538182573511, 2.42035122585773161268303658130, 2.94234495738880880291176160224, 3.54100438124106734535074497064, 3.67769329628755774515261565780, 3.90193060525851594972408428590, 3.94955173303974419353559382745, 4.66220408144396368763745782097, 4.92203804438125853298384469613, 5.26546767610340386199346250864, 5.37956984663547349741553726609, 5.82439314737636279219987513629, 5.83241277213078958917455451227, 6.12751598329163930926402689935, 6.63255680361938415060013258247, 6.75151195981889867314297622062, 6.79708566431453852598271358562, 6.89258776389101913633102092229, 7.38226228244397334552178217688, 7.60567950771521789978397705076, 7.61490590206857053901811644296
