| L(s) = 1 | + 9-s − 10·25-s + 14·49-s + 4·79-s − 8·81-s + 44·109-s − 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 10·225-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 2·25-s + 2·49-s + 0.450·79-s − 8/9·81-s + 4.21·109-s − 2.36·121-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 − 2.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s − 2/3·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1733273477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1733273477\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 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 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 149 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.86236835312413898664527206301, −6.17222088708926379418825205322, −6.09812943164351163838147747091, −6.07152747577467004147520299713, −6.02149240353965014746300686090, −5.69808629692375829072147900302, −5.31712709551524249238507716730, −5.05159281374872338844494724729, −4.97478162678790030623157479396, −4.75683133403135700089606486296, −4.52897198952366180722656194167, −4.07953120959303464255623529220, −3.82129032437384323329234736002, −3.81300495629830335409044942467, −3.81058416471280730985409857525, −3.21284286801102683484996634165, −3.00067098886229579902133289545, −2.73555401149597140367336528649, −2.25620305537984466720236272901, −2.22092022367522043124687045394, −2.00229986031107690271895440234, −1.50504930325271593128499519220, −1.06095138631415607151277542737, −0.987091973113866678190838040365, −0.07773403085298341855658311822,
0.07773403085298341855658311822, 0.987091973113866678190838040365, 1.06095138631415607151277542737, 1.50504930325271593128499519220, 2.00229986031107690271895440234, 2.22092022367522043124687045394, 2.25620305537984466720236272901, 2.73555401149597140367336528649, 3.00067098886229579902133289545, 3.21284286801102683484996634165, 3.81058416471280730985409857525, 3.81300495629830335409044942467, 3.82129032437384323329234736002, 4.07953120959303464255623529220, 4.52897198952366180722656194167, 4.75683133403135700089606486296, 4.97478162678790030623157479396, 5.05159281374872338844494724729, 5.31712709551524249238507716730, 5.69808629692375829072147900302, 6.02149240353965014746300686090, 6.07152747577467004147520299713, 6.09812943164351163838147747091, 6.17222088708926379418825205322, 6.86236835312413898664527206301
