| L(s) = 1 | − 8·7-s − 2·16-s − 8·23-s + 32·37-s + 32·43-s + 32·49-s − 32·53-s + 16·67-s − 16·71-s + 16·107-s + 16·112-s + 32·113-s − 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 1/2·16-s − 1.66·23-s + 5.26·37-s + 4.87·43-s + 32/7·49-s − 4.39·53-s + 1.95·67-s − 1.89·71-s + 1.54·107-s + 1.51·112-s + 3.01·113-s − 5.09·121-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 + 5.04·161-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3411163134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3411163134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 - 48 T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 - 8 T^{2} + 32 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 240 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 252 T^{4} + 17030 T^{8} - 252 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 24 T^{2} + 288 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 4 T + 8 T^{2} + 84 T^{3} + 878 T^{4} + 84 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 108 T^{2} + 4806 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 16 T + 128 T^{2} - 1040 T^{3} + 7666 T^{4} - 1040 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 148 T^{2} + 8806 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 + 6020 T^{4} + 17872774 T^{8} + 6020 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 16 T + 128 T^{2} + 784 T^{3} + 4786 T^{4} + 784 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 136 T^{2} + 10336 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 96 T^{2} + 8688 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8 T + 32 T^{2} + 552 T^{3} - 8974 T^{4} + 552 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 1220 T^{4} + 55730374 T^{8} + 1220 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 208 T^{2} + 22498 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 4224 T^{4} + 99283874 T^{8} - 4224 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 60 T^{2} + 10470 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 11900 T^{4} + 108781062 T^{8} - 11900 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−7.15435746260915370456338921797, −6.65806377191328104629730671593, −6.57503830327895080777047985474, −6.56757197148260786786250918424, −6.47683998663292806679556344576, −6.12279359553086067769335308622, −6.04506168858075457189373479891, −5.83330329117595111325775970775, −5.80269736865371601942855919682, −5.72631884677360801430512556421, −5.34775913171713685437555407601, −4.88199365542020945463572457055, −4.64317506760503177735883080044, −4.61186103028457701914632051657, −4.34714517064839696655961222057, −4.07963943454371272144407811349, −3.96264512066317996909061166558, −3.58870751755677526487013057970, −3.58589607845968787871267683122, −2.95982849167023176394994439581, −2.93893812303338789770631387506, −2.58592318576753850291609497310, −2.49398922054417643061211859424, −2.11189885120549725243992584967, −1.09516995505325869631902976096,
1.09516995505325869631902976096, 2.11189885120549725243992584967, 2.49398922054417643061211859424, 2.58592318576753850291609497310, 2.93893812303338789770631387506, 2.95982849167023176394994439581, 3.58589607845968787871267683122, 3.58870751755677526487013057970, 3.96264512066317996909061166558, 4.07963943454371272144407811349, 4.34714517064839696655961222057, 4.61186103028457701914632051657, 4.64317506760503177735883080044, 4.88199365542020945463572457055, 5.34775913171713685437555407601, 5.72631884677360801430512556421, 5.80269736865371601942855919682, 5.83330329117595111325775970775, 6.04506168858075457189373479891, 6.12279359553086067769335308622, 6.47683998663292806679556344576, 6.56757197148260786786250918424, 6.57503830327895080777047985474, 6.65806377191328104629730671593, 7.15435746260915370456338921797
Plot not available for L-functions of degree greater than 10.
