| L(s) = 1 | + 8·5-s − 4·9-s + 36·25-s + 8·37-s − 32·45-s − 20·49-s + 40·53-s + 10·81-s − 16·89-s + 8·97-s + 88·113-s + 36·121-s + 120·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 68·169-s + 173-s + 179-s + 181-s + 64·185-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 4/3·9-s + 36/5·25-s + 1.31·37-s − 4.77·45-s − 2.85·49-s + 5.49·53-s + 10/9·81-s − 1.69·89-s + 0.812·97-s + 8.27·113-s + 3.27·121-s + 10.7·125-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 + 5.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 4.70·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.96143315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.96143315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 5 | \( ( 1 - T )^{8} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 10 T^{2} + 58 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 34 T^{2} + 562 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 50 T^{2} + 1138 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 40 T^{2} + 1198 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 48 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 32 T^{2} - 542 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 154 T^{2} + 9562 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 56 T^{2} + 1246 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T^{2} - 1882 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 120 T^{2} + 10238 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 152 T^{2} + 15598 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 50 T^{2} - 3342 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 148 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 274 T^{2} + 31962 T^{4} + 274 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.60241940691337664756339118217, −3.57846592012896226064113728682, −3.38763652850889313629775705886, −3.19692656623676055493111689315, −3.14379964922152792643200760286, −3.02756629349794607707163487653, −3.01521114200887218548432875308, −2.81634523312050771679740027660, −2.60185968875867154312336421458, −2.57396483711030138302103505231, −2.41727817557103050911643753240, −2.21195431249332536310847064964, −2.13413428839076354756468100198, −2.08616407648655303914781932786, −2.07524902223853072074923319351, −1.88993409025940680727719968922, −1.76759714115748091924669606319, −1.42705030247846877707521834680, −1.33166229023194234486146826881, −1.17468005716145731613494875714, −1.05874062218772185016789856702, −0.836343177969994225917052606043, −0.74895398931156652786746570772, −0.40309120757923529974783926131, −0.27312590610215048942166256084,
0.27312590610215048942166256084, 0.40309120757923529974783926131, 0.74895398931156652786746570772, 0.836343177969994225917052606043, 1.05874062218772185016789856702, 1.17468005716145731613494875714, 1.33166229023194234486146826881, 1.42705030247846877707521834680, 1.76759714115748091924669606319, 1.88993409025940680727719968922, 2.07524902223853072074923319351, 2.08616407648655303914781932786, 2.13413428839076354756468100198, 2.21195431249332536310847064964, 2.41727817557103050911643753240, 2.57396483711030138302103505231, 2.60185968875867154312336421458, 2.81634523312050771679740027660, 3.01521114200887218548432875308, 3.02756629349794607707163487653, 3.14379964922152792643200760286, 3.19692656623676055493111689315, 3.38763652850889313629775705886, 3.57846592012896226064113728682, 3.60241940691337664756339118217
Plot not available for L-functions of degree greater than 10.
