| L(s) = 1 | + 4·4-s − 16·29-s + 224·41-s − 192·49-s + 256·61-s − 176·89-s − 656·101-s + 256·109-s − 64·116-s + 168·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 896·164-s + 167-s + 1.20e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 768·196-s + ⋯ |
| L(s) = 1 | + 4-s − 0.551·29-s + 5.46·41-s − 3.91·49-s + 4.19·61-s − 1.97·89-s − 6.49·101-s + 2.34·109-s − 0.551·116-s + 1.38·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 5.46·164-s + 0.00598·167-s + 7.14·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s − 3.91·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.374791450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374791450\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T^{2} + p^{4} T^{4} - p^{6} T^{6} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 96 T^{2} + 6606 T^{4} + 96 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 84 T^{2} - 7674 T^{4} - 84 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 604 T^{2} + 147046 T^{4} - 604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 444 T^{2} + 170246 T^{4} - 444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 1124 T^{2} + 571366 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 1856 T^{2} + 1404046 T^{4} + 1856 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 1606 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1524 T^{2} + 1236966 T^{4} - 1524 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4444 T^{2} + 8653606 T^{4} - 4444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 56 T + 3966 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 6896 T^{2} + 18665806 T^{4} + 6896 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 4736 T^{2} + 14725966 T^{4} + 4736 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 6364 T^{2} + 22034086 T^{4} - 6364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 8164 T^{2} + 34261926 T^{4} - 8164 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 64 T + 5086 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 7536 T^{2} + 47276046 T^{4} + 7536 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12084 T^{2} + 81146406 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 2364 T^{2} - 31017274 T^{4} - 2364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 11844 T^{2} + 72414726 T^{4} - 11844 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 21296 T^{2} + 201120526 T^{4} + 21296 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 44 T + 8326 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 27484 T^{2} + 353355846 T^{4} - 27484 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.19625424913180187149509191638, −4.07126010620538566974582117213, −3.94533233315345884430363816723, −3.67798752104931594096297049205, −3.61385669398416320054661965834, −3.40031763408424442854448448245, −3.28526335484262312061227001212, −3.07990334521480666752596870686, −2.97819924861037966761337921555, −2.78683226327264713746826759168, −2.72029458406553535689003061424, −2.68885360663636683419919397043, −2.44899298569836810492336082234, −2.23409898944043421477932975678, −2.08383745098272626407257036812, −1.99623585585548448141247441407, −1.74671266021176587676445761238, −1.66945626997710078208796103136, −1.54720723339663861218119057307, −1.18390166086736544379095978964, −0.925523787964879245609929429359, −0.847770143156409966253232548996, −0.71014231667832296768451325593, −0.44305415883779527692579486907, −0.07171840508930801372351664917,
0.07171840508930801372351664917, 0.44305415883779527692579486907, 0.71014231667832296768451325593, 0.847770143156409966253232548996, 0.925523787964879245609929429359, 1.18390166086736544379095978964, 1.54720723339663861218119057307, 1.66945626997710078208796103136, 1.74671266021176587676445761238, 1.99623585585548448141247441407, 2.08383745098272626407257036812, 2.23409898944043421477932975678, 2.44899298569836810492336082234, 2.68885360663636683419919397043, 2.72029458406553535689003061424, 2.78683226327264713746826759168, 2.97819924861037966761337921555, 3.07990334521480666752596870686, 3.28526335484262312061227001212, 3.40031763408424442854448448245, 3.61385669398416320054661965834, 3.67798752104931594096297049205, 3.94533233315345884430363816723, 4.07126010620538566974582117213, 4.19625424913180187149509191638
Plot not available for L-functions of degree greater than 10.
