| L(s) = 1 | + 4·9-s + 4·25-s − 24·29-s − 16·49-s − 8·61-s + 16·79-s + 16·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 4/5·25-s − 4.45·29-s − 2.28·49-s − 1.02·61-s + 1.80·79-s + 16/9·81-s + 4·121-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 + 4/13·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693959994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693959994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 2 T^{2} - 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 8 T^{2} + 30 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 p T^{2} + 342 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 28 T^{2} + 438 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 46 T^{2} + 1062 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 58 T^{2} + 1710 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 22 T^{2} + 342 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 80 T^{2} + 3582 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 76 T^{2} + 4470 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 106 T^{2} + 6318 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 56 T^{2} + 4446 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 172 T^{2} + 12678 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 46 T^{2} + 7302 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 200 T^{2} + 18222 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 238 T^{2} + 23718 T^{4} - 238 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 224 T^{2} + 22446 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 56 T^{2} - 4338 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 236 T^{2} + 27366 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−5.49405522674696637756099548560, −5.14888512411359633695121673671, −5.14106146244585628738676197479, −4.82721571351949580683652488741, −4.73355821031781078324458696033, −4.69462248036237674970182723789, −4.57132868850683820827067915798, −4.52780398632788205743878440926, −4.03780666551540051716002148793, −3.90613938289059777092992060759, −3.85557078244062818850473660291, −3.60092490133216161012012192919, −3.51061612675524548810961982939, −3.40129154430712144561358409556, −3.20968770176302602339701251735, −3.07030047949319203839179817174, −2.57360796458670893059355337453, −2.51728384451142389207985567391, −2.17168941959692474062811147196, −2.11739811192374868269543172472, −1.74778547814625556394771400359, −1.54954256022427537684495640079, −1.52491595230567678578685632686, −0.956121926463777211496794486991, −0.39571677344501970876545734483,
0.39571677344501970876545734483, 0.956121926463777211496794486991, 1.52491595230567678578685632686, 1.54954256022427537684495640079, 1.74778547814625556394771400359, 2.11739811192374868269543172472, 2.17168941959692474062811147196, 2.51728384451142389207985567391, 2.57360796458670893059355337453, 3.07030047949319203839179817174, 3.20968770176302602339701251735, 3.40129154430712144561358409556, 3.51061612675524548810961982939, 3.60092490133216161012012192919, 3.85557078244062818850473660291, 3.90613938289059777092992060759, 4.03780666551540051716002148793, 4.52780398632788205743878440926, 4.57132868850683820827067915798, 4.69462248036237674970182723789, 4.73355821031781078324458696033, 4.82721571351949580683652488741, 5.14106146244585628738676197479, 5.14888512411359633695121673671, 5.49405522674696637756099548560
Plot not available for L-functions of degree greater than 10.
