| L(s) = 1 | − 48·13-s − 8·25-s − 4·49-s − 48·61-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.19e3·169-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 + ⋯ |
| L(s) = 1 | − 13.3·13-s − 8/5·25-s − 4/7·49-s − 6.14·61-s + 8/11·121-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 + 91.6·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01272082621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01272082621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
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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
−3.37011719652534080092856633296, −3.30733693801874088062718307759, −3.11127485072595814664069091424, −3.01769987649798842576989762904, −2.91987525360664458311832583820, −2.83309069221152600399256438979, −2.60040169621516099104274956635, −2.45952938677274292317817405377, −2.45566592114161714118012202897, −2.44842353426883321576498522601, −2.41445287808050129552642103096, −2.40089517523627754296147800157, −2.21674038826987141264034174774, −1.98281573854961590577186828808, −1.98039850215446427567845861289, −1.72916565447866381577097574503, −1.59561145858859799287572830157, −1.38978326280876232626119412697, −1.25537338920871280979528818392, −1.20817323794339089206586582713, −0.57456780150481475725401925302, −0.45504247949776373646925496920, −0.24044999412707426861442947642, −0.20826799258436819791606386066, −0.05397370466575565446371796296,
0.05397370466575565446371796296, 0.20826799258436819791606386066, 0.24044999412707426861442947642, 0.45504247949776373646925496920, 0.57456780150481475725401925302, 1.20817323794339089206586582713, 1.25537338920871280979528818392, 1.38978326280876232626119412697, 1.59561145858859799287572830157, 1.72916565447866381577097574503, 1.98039850215446427567845861289, 1.98281573854961590577186828808, 2.21674038826987141264034174774, 2.40089517523627754296147800157, 2.41445287808050129552642103096, 2.44842353426883321576498522601, 2.45566592114161714118012202897, 2.45952938677274292317817405377, 2.60040169621516099104274956635, 2.83309069221152600399256438979, 2.91987525360664458311832583820, 3.01769987649798842576989762904, 3.11127485072595814664069091424, 3.30733693801874088062718307759, 3.37011719652534080092856633296
Plot not available for L-functions of degree greater than 10.
