| L(s) = 1 | + 8·11-s − 8·37-s − 48·49-s − 16·59-s + 24·61-s − 72·83-s − 64·97-s + 80·107-s + 48·109-s + 32·121-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 + ⋯ |
| L(s) = 1 | + 2.41·11-s − 1.31·37-s − 6.85·49-s − 2.08·59-s + 3.07·61-s − 7.90·83-s − 6.49·97-s + 7.73·107-s + 4.59·109-s + 2.90·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.695959097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.695959097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4 T^{4} - 474 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 4 T + 8 T^{2} - 28 T^{3} + 82 T^{4} - 28 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 40 T^{2} + 786 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 292 T^{4} - 25242 T^{8} + 292 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 - 964 T^{4} + 566886 T^{8} - 964 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 40 T^{2} + 594 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 8 T^{2} + 60 T^{3} - 34 T^{4} + 60 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 40 T^{2} + 2034 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 3548 T^{4} + 6433446 T^{8} - 3548 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 - 4868 T^{4} + 16478118 T^{8} - 4868 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 8 T + 32 T^{2} - 232 T^{3} - 6062 T^{4} - 232 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 10276 T^{4} + 56007654 T^{8} + 10276 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 188 T^{2} + 18726 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 168 T^{2} + 14738 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 36 T + 648 T^{2} + 8604 T^{3} + 89906 T^{4} + 8604 p T^{5} + 648 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 80 T^{2} + 15714 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.86374330963046163264406552577, −3.60458974375386289454254272459, −3.36891718852541771640345142442, −3.36797086843620585146418347441, −3.34335663361429085279507209367, −3.17730336109823957533461250076, −3.14984867721440888342653711877, −2.88658762730873834587017270637, −2.86358799698313608502369173212, −2.71357309489454288585204414094, −2.56203516893676418359011319053, −2.47224732705584523056751745979, −2.07964706046988615092313500018, −2.03074245774891397491230967183, −1.81685007596041001294487925638, −1.77878901129439170370872979566, −1.66601025763729688492218319363, −1.58022313911352771284523578129, −1.39660570972588933576920314954, −1.23580200964198419393432036259, −1.14851935025485350644863021976, −0.912786107129840052713000431697, −0.49313189180522545684037689314, −0.32562356311971391279159552432, −0.25774047813766212183726541921,
0.25774047813766212183726541921, 0.32562356311971391279159552432, 0.49313189180522545684037689314, 0.912786107129840052713000431697, 1.14851935025485350644863021976, 1.23580200964198419393432036259, 1.39660570972588933576920314954, 1.58022313911352771284523578129, 1.66601025763729688492218319363, 1.77878901129439170370872979566, 1.81685007596041001294487925638, 2.03074245774891397491230967183, 2.07964706046988615092313500018, 2.47224732705584523056751745979, 2.56203516893676418359011319053, 2.71357309489454288585204414094, 2.86358799698313608502369173212, 2.88658762730873834587017270637, 3.14984867721440888342653711877, 3.17730336109823957533461250076, 3.34335663361429085279507209367, 3.36797086843620585146418347441, 3.36891718852541771640345142442, 3.60458974375386289454254272459, 3.86374330963046163264406552577
Plot not available for L-functions of degree greater than 10.
