| L(s) = 1 | − 28·49-s − 20·81-s + 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 4·49-s − 2.22·81-s + 8·121-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 + 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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736817555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736817555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 + 10 T^{4} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 22 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 - 310 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - p T^{2} )^{8} \) |
| 19 | \( ( 1 + 650 T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( ( 1 + p T^{2} )^{8} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 - p T^{2} )^{8} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 + 1130 T^{4} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 7370 T^{4} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - p T^{2} )^{8} \) |
| 71 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 13130 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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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
−3.99112206991323480395948132801, −3.83374050397807626923635518839, −3.57420121635516214827238509819, −3.31049570588118957232065995648, −3.28765269988287909930543923491, −3.27192836174525397274491024192, −3.21465250860703987468641837063, −3.13378362240551053303312175677, −3.04783527168035286117061215094, −2.79290465102136854393894052598, −2.77878772948080805674055621091, −2.38282262187511735976021785188, −2.30928586644260383598414412713, −2.13886816851499135542650551143, −2.00501283909746227249308836956, −1.79697438066615160203114329376, −1.79543162845237260553689733651, −1.75761622353604833143890130413, −1.61938729458132757213472895836, −1.07056485266897956125749857677, −1.04628088297666194502980030677, −0.986317556213917886030009066632, −0.57532414993034553444522376973, −0.50120665552685616144983625922, −0.12993200639391534329578068949,
0.12993200639391534329578068949, 0.50120665552685616144983625922, 0.57532414993034553444522376973, 0.986317556213917886030009066632, 1.04628088297666194502980030677, 1.07056485266897956125749857677, 1.61938729458132757213472895836, 1.75761622353604833143890130413, 1.79543162845237260553689733651, 1.79697438066615160203114329376, 2.00501283909746227249308836956, 2.13886816851499135542650551143, 2.30928586644260383598414412713, 2.38282262187511735976021785188, 2.77878772948080805674055621091, 2.79290465102136854393894052598, 3.04783527168035286117061215094, 3.13378362240551053303312175677, 3.21465250860703987468641837063, 3.27192836174525397274491024192, 3.28765269988287909930543923491, 3.31049570588118957232065995648, 3.57420121635516214827238509819, 3.83374050397807626923635518839, 3.99112206991323480395948132801
Plot not available for L-functions of degree greater than 10.
