| L(s) = 1 | + 4·4-s − 8·9-s − 208·13-s + 26·16-s + 80·25-s − 32·36-s − 128·43-s − 252·49-s − 832·52-s + 184·61-s + 32·64-s + 80·79-s + 178·81-s + 320·100-s − 976·103-s + 1.66e3·117-s + 884·121-s + 127-s + 131-s + 137-s + 139-s − 208·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 4-s − 8/9·9-s − 16·13-s + 13/8·16-s + 16/5·25-s − 8/9·36-s − 2.97·43-s − 5.14·49-s − 16·52-s + 3.01·61-s + 1/2·64-s + 1.01·79-s + 2.19·81-s + 16/5·100-s − 9.47·103-s + 14.2·117-s + 7.30·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.44·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.002241445408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002241445408\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + 4 T^{2} - 65 T^{4} + 4 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 18 p T^{2} + 157 p^{2} T^{4} + 18 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + p T )^{16} \) |
| good | 2 | \( ( 1 - p T^{2} - 7 T^{4} + 21 p T^{6} - 199 T^{8} + 21 p^{5} T^{10} - 7 p^{8} T^{12} - p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 5 | \( ( 1 - 8 p T^{2} + 742 T^{4} + 3136 p T^{6} - 654461 T^{8} + 3136 p^{5} T^{10} + 742 p^{8} T^{12} - 8 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 11 | \( ( 1 - 442 T^{2} + 117439 T^{4} - 21500206 T^{6} + 2997622420 T^{8} - 21500206 p^{4} T^{10} + 117439 p^{8} T^{12} - 442 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 + 302 T^{2} + 9161 T^{4} - 25669698 T^{6} - 7862152876 T^{8} - 25669698 p^{4} T^{10} + 9161 p^{8} T^{12} + 302 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 + 1374 T^{2} + 1156343 T^{4} + 647004234 T^{6} + 272127078852 T^{8} + 647004234 p^{4} T^{10} + 1156343 p^{8} T^{12} + 1374 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 23 | \( ( 1 + 1052 T^{2} + 546314 T^{4} + 744816 T^{6} - 73975187245 T^{8} + 744816 p^{4} T^{10} + 546314 p^{8} T^{12} + 1052 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 - 1418 T^{2} + 1641275 T^{4} - 1418 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 31 | \( ( 1 + 3480 T^{2} + 7240070 T^{4} + 10521042240 T^{6} + 11683014553059 T^{8} + 10521042240 p^{4} T^{10} + 7240070 p^{8} T^{12} + 3480 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 37 | \( ( 1 + 4356 T^{2} + 10758698 T^{4} + 19461370896 T^{6} + 28429090767027 T^{8} + 19461370896 p^{4} T^{10} + 10758698 p^{8} T^{12} + 4356 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 + 4196 T^{2} + 9101318 T^{4} + 4196 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 43 | \( ( 1 + 16 T + 3740 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{8} \) |
| 47 | \( ( 1 - 3974 T^{2} + 5729423 T^{4} - 1207662834 T^{6} + 394553263892 T^{8} - 1207662834 p^{4} T^{10} + 5729423 p^{8} T^{12} - 3974 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 - 1490 T^{2} - 10972439 T^{4} + 3856750270 T^{6} + 94265737987060 T^{8} + 3856750270 p^{4} T^{10} - 10972439 p^{8} T^{12} - 1490 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 - 12314 T^{2} + 89752607 T^{4} - 463588445838 T^{6} + 1845138985170644 T^{8} - 463588445838 p^{4} T^{10} + 89752607 p^{8} T^{12} - 12314 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 - 46 T - 4073 T^{2} + 57638 T^{3} + 16584244 T^{4} + 57638 p^{2} T^{5} - 4073 p^{4} T^{6} - 46 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 67 | \( ( 1 + 7470 T^{2} + 28120055 T^{4} - 94281835590 T^{6} - 764289185451516 T^{8} - 94281835590 p^{4} T^{10} + 28120055 p^{8} T^{12} + 7470 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 + 10426 T^{2} + 61231893 T^{4} + 10426 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 73 | \( ( 1 - 104 T^{2} - 56439098 T^{4} + 36043072 T^{6} + 2379825543338659 T^{8} + 36043072 p^{4} T^{10} - 56439098 p^{8} T^{12} - 104 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 - 20 T - 9982 T^{2} + 42000 T^{3} + 66339443 T^{4} + 42000 p^{2} T^{5} - 9982 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 16552 T^{2} + 134515866 T^{4} + 16552 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 89 | \( ( 1 - 268 T^{2} - 60648182 T^{4} + 17356879568 T^{6} - 254176075779629 T^{8} + 17356879568 p^{4} T^{10} - 60648182 p^{8} T^{12} - 268 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 97 | \( ( 1 - 36348 T^{2} + 507335590 T^{4} - 36348 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.02344362601210208119520393852, −2.99673902624545950700409348764, −2.96846380773337638461589843288, −2.93381652437885093446290023516, −2.69055752671978665171910562493, −2.52776880852218163536268424801, −2.51509650595855706194163173081, −2.42345411902072511851381860115, −2.25707380329880714943159680599, −2.21175625546058801077049749773, −2.20506550686710427432610259517, −2.08500795513678458203654541430, −1.90752719284588504260447447401, −1.85752459486927191815968695005, −1.84004976033896518960034342342, −1.75062133700175812823845361124, −1.72564292529057497248116329466, −1.30866934740735484702880723715, −0.854638644818097353664601628017, −0.814999746596776804421839893438, −0.59843072560089989222241211480, −0.51228019158960999593241621852, −0.42205917276074737879089976426, −0.13014537847348308825329840861, −0.01373643751722746179458149861,
0.01373643751722746179458149861, 0.13014537847348308825329840861, 0.42205917276074737879089976426, 0.51228019158960999593241621852, 0.59843072560089989222241211480, 0.814999746596776804421839893438, 0.854638644818097353664601628017, 1.30866934740735484702880723715, 1.72564292529057497248116329466, 1.75062133700175812823845361124, 1.84004976033896518960034342342, 1.85752459486927191815968695005, 1.90752719284588504260447447401, 2.08500795513678458203654541430, 2.20506550686710427432610259517, 2.21175625546058801077049749773, 2.25707380329880714943159680599, 2.42345411902072511851381860115, 2.51509650595855706194163173081, 2.52776880852218163536268424801, 2.69055752671978665171910562493, 2.93381652437885093446290023516, 2.96846380773337638461589843288, 2.99673902624545950700409348764, 3.02344362601210208119520393852
Plot not available for L-functions of degree greater than 10.
