| L(s) = 1 | − 16·4-s − 52·7-s − 292·13-s + 192·16-s + 740·19-s + 2.03e3·25-s + 832·28-s − 5.05e3·31-s + 4.42e3·37-s + 428·43-s − 7.42e3·49-s + 4.67e3·52-s + 4.49e3·61-s − 2.04e3·64-s + 1.64e4·67-s − 2.08e4·73-s − 1.18e4·76-s − 1.67e4·79-s + 1.51e4·91-s − 3.80e3·97-s − 3.25e4·100-s + 1.13e4·103-s + 9.11e3·109-s − 9.98e3·112-s + 4.59e4·121-s + 8.08e4·124-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 1.06·7-s − 1.72·13-s + 3/4·16-s + 2.04·19-s + 3.25·25-s + 1.06·28-s − 5.26·31-s + 3.23·37-s + 0.231·43-s − 3.09·49-s + 1.72·52-s + 1.20·61-s − 1/2·64-s + 3.66·67-s − 3.90·73-s − 2.04·76-s − 2.68·79-s + 1.83·91-s − 0.404·97-s − 3.25·100-s + 1.07·103-s + 0.767·109-s − 0.795·112-s + 3.13·121-s + 5.26·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7204681862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7204681862\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2032 T^{2} + 1758831 T^{4} - 2032 p^{8} T^{6} + p^{16} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 26 T + 4728 T^{2} + 26 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 45928 T^{2} + 7807650 p^{2} T^{4} - 45928 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 146 T + 38151 T^{2} + 146 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 104 T^{2} + 13445531079 T^{4} + 104 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 370 T + 61344 T^{2} - 370 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 657664 T^{2} + 236408934786 T^{4} - 657664 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1983160 T^{2} + 1983655205799 T^{4} - 1983160 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 2528 T + 3366006 T^{2} + 2528 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 2212 T + 3297531 T^{2} - 2212 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 954232 T^{2} + 15817350484626 T^{4} - 954232 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 214 T + 2017968 T^{2} - 214 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 5282452 T^{2} + 45013093556406 T^{4} - 5282452 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 14240632 T^{2} + 170431500573906 T^{4} - 14240632 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 30153076 T^{2} + 519632924456886 T^{4} - 30153076 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 2248 T + 20274855 T^{2} - 2248 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 8218 T + 53142360 T^{2} - 8218 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 94710496 T^{2} + 700726549794 p^{2} T^{4} - 94710496 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 10400 T + 83729319 T^{2} + 10400 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 8366 T + 48566448 T^{2} + 8366 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 33630580 T^{2} + 256606886767974 T^{4} - 33630580 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 216095728 T^{2} + 19466598550909983 T^{4} - 216095728 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 1904 T + 35383158 T^{2} + 1904 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−8.810996409960056847808639185354, −8.496668899540929299296503347407, −8.261856542059735469636612436860, −7.75321761937205607032169099935, −7.51261649846480470313988713622, −7.29145816747919170539095026205, −7.18292307525440879386018873701, −6.79419188476351437326695673882, −6.57991411437251085272654047654, −5.95907384071218313622818984009, −5.78912348445818258873716298890, −5.32992561418345940617728327200, −5.29115893671420597734281118525, −4.90504856023796474841021919004, −4.52416161438298825785160401146, −4.36337563328477288560601507045, −3.56876772560636869069034168734, −3.54517972732848295191979080730, −3.16285390029127200815961241499, −2.72186972206134983475403718844, −2.48172178819455486474809005319, −1.66675364723586715170553925647, −1.26609845041106691451725926704, −0.66628777510638306384980803902, −0.21188320441986089515791194891,
0.21188320441986089515791194891, 0.66628777510638306384980803902, 1.26609845041106691451725926704, 1.66675364723586715170553925647, 2.48172178819455486474809005319, 2.72186972206134983475403718844, 3.16285390029127200815961241499, 3.54517972732848295191979080730, 3.56876772560636869069034168734, 4.36337563328477288560601507045, 4.52416161438298825785160401146, 4.90504856023796474841021919004, 5.29115893671420597734281118525, 5.32992561418345940617728327200, 5.78912348445818258873716298890, 5.95907384071218313622818984009, 6.57991411437251085272654047654, 6.79419188476351437326695673882, 7.18292307525440879386018873701, 7.29145816747919170539095026205, 7.51261649846480470313988713622, 7.75321761937205607032169099935, 8.261856542059735469636612436860, 8.496668899540929299296503347407, 8.810996409960056847808639185354
