| L(s) = 1 | + 6·9-s − 120·29-s + 144·41-s − 190·49-s + 140·61-s + 27·81-s − 576·89-s − 192·101-s − 676·109-s + 460·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 434·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 4.13·29-s + 3.51·41-s − 3.87·49-s + 2.29·61-s + 1/3·81-s − 6.47·89-s − 1.90·101-s − 6.20·109-s + 3.80·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.56·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7043789181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7043789181\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 95 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 217 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 542 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 359 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 758 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1895 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 36 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3671 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2390 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 434 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5510 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8111 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1970 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 6814 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 6674 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 8486 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 13943 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.70914653173365013321895008960, −6.59438630692957160981668187443, −6.47211707500753603677775271889, −5.95050031053504605572890103710, −5.76402387187312834536328528913, −5.71748178563081017341005063189, −5.56454754883464187587039666493, −5.10496919745994120992175968115, −5.08090220550570368596620455373, −4.79649580902870168528975995178, −4.36828006469821018332004503201, −4.02223465130128545243532778045, −3.98997114949658085910902153612, −3.88987851725640632245695980432, −3.71797406006943714035799192229, −3.05089525086717588271436051533, −2.96243241309533412565288641995, −2.67817604179921099778019859525, −2.38708443020714735178071088897, −2.04854346431068057268449847962, −1.54973506931067582063636369015, −1.45926930268131202525043035415, −1.30606940633592018898045871650, −0.54216950204474367361884569964, −0.14123767706450036840143239388,
0.14123767706450036840143239388, 0.54216950204474367361884569964, 1.30606940633592018898045871650, 1.45926930268131202525043035415, 1.54973506931067582063636369015, 2.04854346431068057268449847962, 2.38708443020714735178071088897, 2.67817604179921099778019859525, 2.96243241309533412565288641995, 3.05089525086717588271436051533, 3.71797406006943714035799192229, 3.88987851725640632245695980432, 3.98997114949658085910902153612, 4.02223465130128545243532778045, 4.36828006469821018332004503201, 4.79649580902870168528975995178, 5.08090220550570368596620455373, 5.10496919745994120992175968115, 5.56454754883464187587039666493, 5.71748178563081017341005063189, 5.76402387187312834536328528913, 5.95050031053504605572890103710, 6.47211707500753603677775271889, 6.59438630692957160981668187443, 6.70914653173365013321895008960
