| L(s) = 1 | + 2·4-s + 2·9-s − 4·11-s + 16-s + 8·19-s − 8·29-s − 32·31-s + 4·36-s − 8·41-s − 8·44-s + 40·59-s − 12·61-s − 2·64-s − 4·71-s + 16·76-s + 32·79-s + 81-s − 8·89-s − 8·99-s + 88·109-s − 16·116-s + 50·121-s − 64·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4-s + 2/3·9-s − 1.20·11-s + 1/4·16-s + 1.83·19-s − 1.48·29-s − 5.74·31-s + 2/3·36-s − 1.24·41-s − 1.20·44-s + 5.20·59-s − 1.53·61-s − 1/4·64-s − 0.474·71-s + 1.83·76-s + 3.60·79-s + 1/9·81-s − 0.847·89-s − 0.804·99-s + 8.42·109-s − 1.48·116-s + 4.54·121-s − 5.74·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.733189994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.733189994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 14 T^{2} + 27 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 7 | \( ( 1 - 4 T + 8 T^{2} + 24 T^{3} - 97 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )( 1 + 4 T + 8 T^{2} - 24 T^{3} - 97 T^{4} - 24 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( 1 + 40 T^{2} + 46 p T^{4} + 9600 T^{6} + 123203 T^{8} + 9600 p^{2} T^{10} + 46 p^{5} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 4 T - 16 T^{2} + 24 T^{3} + 359 T^{4} + 24 p T^{5} - 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 70 T^{2} + 2657 T^{4} + 82950 T^{6} + 2159108 T^{8} + 82950 p^{2} T^{10} + 2657 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 4 T - 6 T^{2} - 144 T^{3} - 821 T^{4} - 144 p T^{5} - 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 8 T + 68 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( 1 + 110 T^{2} + 181 p T^{4} + 293150 T^{6} + 10753348 T^{8} + 293150 p^{2} T^{10} + 181 p^{5} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 4 T - 60 T^{2} - 24 T^{3} + 3439 T^{4} - 24 p T^{5} - 60 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 76 T^{2} + 3927 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 24 T^{2} + 4322 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 6 T - 85 T^{2} - 6 T^{3} + 8724 T^{4} - 6 p T^{5} - 85 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 2 T - 49 T^{2} - 178 T^{3} - 2516 T^{4} - 178 p T^{5} - 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 194 T^{2} + 18627 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 4 T - 76 T^{2} - 344 T^{3} - 881 T^{4} - 344 p T^{5} - 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 306 T^{2} + 51569 T^{4} + 7114194 T^{6} + 800082084 T^{8} + 7114194 p^{2} T^{10} + 51569 p^{4} T^{12} + 306 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.77711848284127417483017648515, −3.63936958021492730524361375262, −3.61466158501532225536387648791, −3.56632553219921607128194909005, −3.29457328807210026436707551534, −3.28903498507064449545022946998, −3.27972287498416944747477379105, −2.93093649607611574082218205114, −2.87120781307994414540673102261, −2.71847764987137620612801562765, −2.65989277407532452562281584630, −2.44201183037425136157062243092, −2.11883696465707211006967032864, −1.95123151662546194562566207789, −1.92530300251845641977946892456, −1.92499625582596355159986409705, −1.91404884859560933572301755164, −1.82081042733772559398909826865, −1.60124901110636547712790806918, −1.20592681270381483151500825509, −1.04205468015676740365987825196, −0.75715584470770128831688486601, −0.63018857481576998454019678405, −0.55565844508511443690418902850, −0.13053961775550309935898707994,
0.13053961775550309935898707994, 0.55565844508511443690418902850, 0.63018857481576998454019678405, 0.75715584470770128831688486601, 1.04205468015676740365987825196, 1.20592681270381483151500825509, 1.60124901110636547712790806918, 1.82081042733772559398909826865, 1.91404884859560933572301755164, 1.92499625582596355159986409705, 1.92530300251845641977946892456, 1.95123151662546194562566207789, 2.11883696465707211006967032864, 2.44201183037425136157062243092, 2.65989277407532452562281584630, 2.71847764987137620612801562765, 2.87120781307994414540673102261, 2.93093649607611574082218205114, 3.27972287498416944747477379105, 3.28903498507064449545022946998, 3.29457328807210026436707551534, 3.56632553219921607128194909005, 3.61466158501532225536387648791, 3.63936958021492730524361375262, 3.77711848284127417483017648515
Plot not available for L-functions of degree greater than 10.
