| L(s) = 1 | − 4·4-s + 12·9-s + 4·16-s + 24·17-s + 20·25-s + 32·29-s − 48·36-s + 12·49-s − 4·53-s + 36·61-s + 16·64-s − 96·68-s + 54·81-s − 80·100-s − 80·113-s − 128·116-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 48·144-s + 149-s + 151-s + 288·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·4-s + 4·9-s + 16-s + 5.82·17-s + 4·25-s + 5.94·29-s − 8·36-s + 12/7·49-s − 0.549·53-s + 4.60·61-s + 2·64-s − 11.6·68-s + 6·81-s − 8·100-s − 7.52·113-s − 11.8·116-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4·144-s + 0.0819·149-s + 0.0813·151-s + 23.2·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.312937925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.312937925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( 1 - 12 T^{2} + 95 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - p T + p T^{2} )^{4}( 1 + p T + p T^{2} )^{4} \) |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2}( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 17 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2}( 1 + 12 T^{2} - 217 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 23 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 36 T^{2} + 335 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + p T^{2} )^{8} \) |
| 43 | \( ( 1 - p T^{2} )^{8} \) |
| 47 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2}( 1 + 68 T^{2} + 2415 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 53 | \( ( 1 + 2 T + p T^{2} )^{4}( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2}( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 61 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | \( ( 1 - 108 T^{2} + p^{2} T^{4} )^{2}( 1 + 108 T^{2} + 7175 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 71 | \( ( 1 - 92 T^{2} + 3423 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + p T^{2} )^{8} \) |
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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
−4.95750438499622312955107709539, −4.92458811890684306135430650472, −4.89285447905024493189007939591, −4.75782826405158674949881228636, −4.74234096511261108875456109388, −4.36507673437716620562888052200, −3.99823932541604244170557268899, −3.97780526825538220387439811533, −3.90263966101640534048127032919, −3.88886281978913435879511542965, −3.67363261330504272808093501771, −3.66516792098991991085072852900, −3.33572461271983129266811446160, −2.95022491527717605826027603483, −2.77422050835587165161161812859, −2.72526682628911051177195760382, −2.69524132836081055243172824055, −2.49186117178390057274819796799, −2.19107044059557812214586906634, −1.35980307354741228475581210534, −1.34470319215366011961181752058, −1.20875945860815920841423373445, −1.11778109995631396784500585946, −1.03282595656565969724810672259, −0.846768546541132471931917638276,
0.846768546541132471931917638276, 1.03282595656565969724810672259, 1.11778109995631396784500585946, 1.20875945860815920841423373445, 1.34470319215366011961181752058, 1.35980307354741228475581210534, 2.19107044059557812214586906634, 2.49186117178390057274819796799, 2.69524132836081055243172824055, 2.72526682628911051177195760382, 2.77422050835587165161161812859, 2.95022491527717605826027603483, 3.33572461271983129266811446160, 3.66516792098991991085072852900, 3.67363261330504272808093501771, 3.88886281978913435879511542965, 3.90263966101640534048127032919, 3.97780526825538220387439811533, 3.99823932541604244170557268899, 4.36507673437716620562888052200, 4.74234096511261108875456109388, 4.75782826405158674949881228636, 4.89285447905024493189007939591, 4.92458811890684306135430650472, 4.95750438499622312955107709539
Plot not available for L-functions of degree greater than 10.
