| L(s) = 1 | + 4·5-s − 10·9-s − 8·17-s − 14·25-s − 40·45-s + 20·49-s − 44·61-s + 32·73-s + 35·81-s − 32·85-s + 58·121-s − 80·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 80·153-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 3.33·9-s − 1.94·17-s − 2.79·25-s − 5.96·45-s + 20/7·49-s − 5.63·61-s + 3.74·73-s + 35/9·81-s − 3.47·85-s + 5.27·121-s − 7.15·125-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 + 6.46·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.023477653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023477653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 16 T^{2} + 718 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 5 T^{2} + 20 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 10 T^{2} + 55 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 29 T^{2} + 448 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 11 T^{2} + 28 p T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + T + p T^{2} )^{8} \) |
| 23 | \( ( 1 - 73 T^{2} + 2352 T^{4} - 73 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 43 T^{2} + 916 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 104 T^{2} + 4558 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 104 T^{2} + 5374 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 120 T^{2} + 6894 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 49 T^{2} + 724 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 53 T^{2} + 4776 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 115 T^{2} + 6676 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 61 T^{2} + 5644 T^{4} + 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 11 T + 114 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 173 T^{2} + 14212 T^{4} + 173 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 160 T^{2} + 13150 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 145 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 72 T^{2} + 10446 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 96 T^{2} + 15470 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 124 T^{2} + 12886 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 96 T^{2} + 1470 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.12929586865785206250528241022, −4.09335871624689699623579309510, −3.83877029815780430779955792941, −3.72931275975511294484546323776, −3.56804049861949522846247980347, −3.50190986633198472985929588709, −3.27996004811677026887283587861, −3.21718395778141571250310120246, −3.14054315114138498032335974684, −2.86408456586488144597702156272, −2.71717189236734116926929378790, −2.60327592727165964909293369584, −2.50888245606522198657795128554, −2.31010804065093342620906739905, −2.29062257275408794050755349972, −2.19666525930915372634763662125, −1.82170974936525142280637151218, −1.77757782025173532871210799920, −1.77250422970883646264766990988, −1.51556262701591832727803067715, −1.31255685127914307311767126837, −0.914329189675644567367615311382, −0.51534631518610585061875270151, −0.37599194479412328489182952397, −0.27093893868677562968284057942,
0.27093893868677562968284057942, 0.37599194479412328489182952397, 0.51534631518610585061875270151, 0.914329189675644567367615311382, 1.31255685127914307311767126837, 1.51556262701591832727803067715, 1.77250422970883646264766990988, 1.77757782025173532871210799920, 1.82170974936525142280637151218, 2.19666525930915372634763662125, 2.29062257275408794050755349972, 2.31010804065093342620906739905, 2.50888245606522198657795128554, 2.60327592727165964909293369584, 2.71717189236734116926929378790, 2.86408456586488144597702156272, 3.14054315114138498032335974684, 3.21718395778141571250310120246, 3.27996004811677026887283587861, 3.50190986633198472985929588709, 3.56804049861949522846247980347, 3.72931275975511294484546323776, 3.83877029815780430779955792941, 4.09335871624689699623579309510, 4.12929586865785206250528241022
Plot not available for L-functions of degree greater than 10.
