| L(s) = 1 | − 52·4-s + 274·9-s + 356·11-s + 1.51e3·16-s − 764·29-s − 1.42e4·36-s − 1.85e4·44-s − 98·49-s − 3.03e4·64-s + 1.78e4·71-s − 2.22e4·79-s + 4.31e4·81-s + 9.75e4·99-s − 4.92e4·109-s + 3.97e4·116-s + 2.06e4·121-s + 127-s + 131-s + 137-s + 139-s + 4.15e5·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.14e5·169-s + ⋯ |
| L(s) = 1 | − 3.25·4-s + 3.38·9-s + 2.94·11-s + 5.92·16-s − 0.908·29-s − 10.9·36-s − 9.56·44-s − 0.0408·49-s − 7.41·64-s + 3.53·71-s − 3.56·79-s + 6.58·81-s + 9.95·99-s − 4.14·109-s + 2.95·116-s + 1.41·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 20.0·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.99·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.858619626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.858619626\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{8} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + 13 p T^{2} + p^{8} T^{4} )^{2} \) |
| 3 | $C_2^2$ | \( ( 1 - 137 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 89 T + p^{4} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 57097 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 68183 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 212042 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 68906 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 191 T + p^{4} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 737642 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 794 p^{2} T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2845078 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6695306 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4941337 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 13261538 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 11092322 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 23947082 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 36108866 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4454 T + p^{4} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 18026018 T^{2} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 5561 T + p^{4} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 90956542 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 124831082 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 91773337 T^{2} + p^{8} T^{4} )^{2} \) |
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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
−8.757295491932914861369857293896, −8.512545804993683661880442588519, −8.169731039020927395243762209300, −7.68571801524337607419092419298, −7.55836283277146200235493668781, −7.39747092358553465036904870529, −6.77954704677749980208412713131, −6.60300056424660057758041764919, −6.53735647500537441695169472276, −6.14389559683088763302086749892, −5.38186072607895411461978962946, −5.21928027558551717417832487207, −5.17673640080607621292351406789, −4.40140020118449282209920252349, −4.35578366080234831855500341990, −4.29050093660580788770182697835, −3.79298568706156423280299952292, −3.71611305188200093592750498106, −3.67762951699722141021849825910, −2.68921841071229965522198422668, −1.82717295568717202683512140379, −1.34548999597018740137104132178, −1.28991936313571238717153188155, −0.931680428209438642526386907378, −0.29477078972629363047125464032,
0.29477078972629363047125464032, 0.931680428209438642526386907378, 1.28991936313571238717153188155, 1.34548999597018740137104132178, 1.82717295568717202683512140379, 2.68921841071229965522198422668, 3.67762951699722141021849825910, 3.71611305188200093592750498106, 3.79298568706156423280299952292, 4.29050093660580788770182697835, 4.35578366080234831855500341990, 4.40140020118449282209920252349, 5.17673640080607621292351406789, 5.21928027558551717417832487207, 5.38186072607895411461978962946, 6.14389559683088763302086749892, 6.53735647500537441695169472276, 6.60300056424660057758041764919, 6.77954704677749980208412713131, 7.39747092358553465036904870529, 7.55836283277146200235493668781, 7.68571801524337607419092419298, 8.169731039020927395243762209300, 8.512545804993683661880442588519, 8.757295491932914861369857293896
