| L(s) = 1 | + 7·4-s − 6·5-s + 84·11-s + 16-s − 112·19-s − 42·20-s + 146·25-s − 636·29-s + 104·31-s + 816·41-s + 588·44-s + 616·49-s − 504·55-s − 372·59-s + 680·61-s + 119·64-s + 72·71-s − 784·76-s − 760·79-s − 6·80-s + 2.23e3·89-s + 672·95-s + 1.02e3·100-s + 2.24e3·101-s + 1.32e3·109-s − 4.45e3·116-s − 176·121-s + ⋯ |
| L(s) = 1 | + 7/8·4-s − 0.536·5-s + 2.30·11-s + 1/64·16-s − 1.35·19-s − 0.469·20-s + 1.16·25-s − 4.07·29-s + 0.602·31-s + 3.10·41-s + 2.01·44-s + 1.79·49-s − 1.23·55-s − 0.820·59-s + 1.42·61-s + 0.232·64-s + 0.120·71-s − 1.18·76-s − 1.08·79-s − 0.00838·80-s + 2.65·89-s + 0.725·95-s + 1.02·100-s + 2.21·101-s + 1.16·109-s − 3.56·116-s − 0.132·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.526949758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.526949758\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 6 T - 22 p T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 7 T^{2} + 3 p^{4} T^{4} - 7 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 88 p T^{2} + 316878 T^{4} - 88 p^{7} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 42 T + 2734 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5008 T^{2} + 13678638 T^{4} - 5008 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12400 T^{2} + 76051038 T^{4} - 12400 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 56 T + 8598 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 46204 T^{2} + 828262758 T^{4} - 46204 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 318 T + 55978 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 52 T + 58782 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 96016 T^{2} + 4637534478 T^{4} - 96016 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 408 T + 177982 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 121900 T^{2} + 14997346998 T^{4} - 121900 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 225580 T^{2} + 31469120358 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 376864 T^{2} + 68697431598 T^{4} - 376864 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 186 T + 419038 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 861340 T^{2} + 346621507638 T^{4} - 861340 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 36 T + 384046 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 429844 T^{2} + 136740794118 T^{4} - 429844 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 380 T + 99678 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 916108 T^{2} + 434315280918 T^{4} - 916108 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1116 T + 1508758 T^{2} - 1116 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2174980 T^{2} + 2500346420358 T^{4} - 2174980 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.38954482900435083956603886387, −11.31818831405434333774152808174, −10.69257677770688124698805160245, −10.61409483093948582862124281204, −10.19387517031200527746003192328, −9.516381846573966986457400854953, −9.222983683466609824833945714863, −9.120732485845489073210119384410, −8.961745637624656302324675940371, −8.281653265664983650020886053974, −8.013020897614252540711249848269, −7.31014132394159855816467660097, −7.28957686173416157598035105732, −7.04765906841444647643904145975, −6.34860308910963547672125058706, −6.21122371265295243962762125332, −5.83349246435197788964917444805, −5.32367951567297272907439055948, −4.50223222199671476751873694112, −4.14107489208732795074021262122, −3.84929438339693226768449064244, −3.33139725979925337978033432666, −2.33154417204343144299479224930, −1.92169493751231708746283251778, −0.845502419906100721065398154315,
0.845502419906100721065398154315, 1.92169493751231708746283251778, 2.33154417204343144299479224930, 3.33139725979925337978033432666, 3.84929438339693226768449064244, 4.14107489208732795074021262122, 4.50223222199671476751873694112, 5.32367951567297272907439055948, 5.83349246435197788964917444805, 6.21122371265295243962762125332, 6.34860308910963547672125058706, 7.04765906841444647643904145975, 7.28957686173416157598035105732, 7.31014132394159855816467660097, 8.013020897614252540711249848269, 8.281653265664983650020886053974, 8.961745637624656302324675940371, 9.120732485845489073210119384410, 9.222983683466609824833945714863, 9.516381846573966986457400854953, 10.19387517031200527746003192328, 10.61409483093948582862124281204, 10.69257677770688124698805160245, 11.31818831405434333774152808174, 11.38954482900435083956603886387
