| L(s) = 1 | − 4·4-s + 4·9-s + 4·11-s + 10·16-s + 32·23-s + 20·25-s − 16·36-s − 8·37-s − 16·44-s + 20·49-s − 56·53-s − 20·64-s − 24·67-s + 16·71-s + 16·81-s − 128·92-s + 16·99-s − 80·100-s − 24·113-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 40·144-s + 32·148-s + 149-s + ⋯ |
| L(s) = 1 | − 2·4-s + 4/3·9-s + 1.20·11-s + 5/2·16-s + 6.67·23-s + 4·25-s − 8/3·36-s − 1.31·37-s − 2.41·44-s + 20/7·49-s − 7.69·53-s − 5/2·64-s − 2.93·67-s + 1.89·71-s + 16/9·81-s − 13.3·92-s + 1.60·99-s − 8·100-s − 2.25·113-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10/3·144-s + 2.63·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542098230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542098230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 2 T^{2} - 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 2 p T^{2} + 54 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 42 T^{2} + 758 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 42 T^{2} + 974 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 29 | \( ( 1 - 56 T^{2} + 1710 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 48 T^{2} + 1154 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 40 T^{2} + 2418 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4 T^{2} + 2358 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 64 T^{2} + 4098 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 14 T + 134 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 226 T^{2} + 19710 T^{4} - 226 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 154 T^{2} + 11670 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 8 T^{2} - 1422 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 82 T^{2} + 13758 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 60 T^{2} + 11318 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−5.97273854245482618408482923496, −5.85554968707577883428354475765, −5.52658940533424529287596662615, −5.43287976001281930911638654443, −5.12777921881293340631834055990, −5.04672329369163560464278041221, −4.94785102320726863390187966568, −4.72637924346504236403334251266, −4.72325137684521762672321986146, −4.60205458637534246654963088967, −4.50850069932062857449365930657, −4.36189856840803510619947827250, −3.92954665589412620179875441003, −3.66316073096595015160724651726, −3.45820939089626915951412972212, −3.35305905845959677044757790650, −3.13191279228066023210381083585, −3.03321474669947362489101445092, −3.01030818027584223214623570382, −2.50826445857973340822604385999, −2.27324442557031176208034093767, −1.46982430814631866928688820913, −1.33009268067362763655028261273, −1.10198934929169141204728050193, −1.05786638568198291507047137292,
1.05786638568198291507047137292, 1.10198934929169141204728050193, 1.33009268067362763655028261273, 1.46982430814631866928688820913, 2.27324442557031176208034093767, 2.50826445857973340822604385999, 3.01030818027584223214623570382, 3.03321474669947362489101445092, 3.13191279228066023210381083585, 3.35305905845959677044757790650, 3.45820939089626915951412972212, 3.66316073096595015160724651726, 3.92954665589412620179875441003, 4.36189856840803510619947827250, 4.50850069932062857449365930657, 4.60205458637534246654963088967, 4.72325137684521762672321986146, 4.72637924346504236403334251266, 4.94785102320726863390187966568, 5.04672329369163560464278041221, 5.12777921881293340631834055990, 5.43287976001281930911638654443, 5.52658940533424529287596662615, 5.85554968707577883428354475765, 5.97273854245482618408482923496
Plot not available for L-functions of degree greater than 10.
