| L(s) = 1 | − 4·3-s + 2·7-s + 19·9-s − 22·11-s − 12·13-s − 90·17-s − 50·19-s − 8·21-s + 24·23-s − 92·27-s − 54·29-s + 24·31-s + 88·33-s + 112·37-s + 48·39-s + 88·41-s + 60·43-s + 4·47-s − 61·49-s + 360·51-s + 200·53-s + 200·57-s − 90·59-s + 46·61-s + 38·63-s + 198·67-s − 96·69-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 2/7·7-s + 19/9·9-s − 2·11-s − 0.923·13-s − 5.29·17-s − 2.63·19-s − 0.380·21-s + 1.04·23-s − 3.40·27-s − 1.86·29-s + 0.774·31-s + 8/3·33-s + 3.02·37-s + 1.23·39-s + 2.14·41-s + 1.39·43-s + 4/47·47-s − 1.24·49-s + 7.05·51-s + 3.77·53-s + 3.50·57-s − 1.52·59-s + 0.754·61-s + 0.603·63-s + 2.95·67-s − 1.39·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1955576024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1955576024\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 12 T + 14 p T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 4 T - p T^{2} + 4 T^{3} + 136 T^{4} + 4 p^{2} T^{5} - p^{5} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 1234 T^{4} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 510 T^{3} + 2816 T^{4} - 510 p^{2} T^{5} + 65 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 p T + 185 T^{2} - 1734 T^{3} - 39712 T^{4} - 1734 p^{2} T^{5} + 185 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 90 T + 3949 T^{2} + 112410 T^{3} + 2256780 T^{4} + 112410 p^{2} T^{5} + 3949 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 50 T + 641 T^{2} - 18162 T^{3} - 700144 T^{4} - 18162 p^{2} T^{5} + 641 p^{4} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T + 937 T^{2} - 17880 T^{3} + 376752 T^{4} - 17880 p^{2} T^{5} + 937 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 54 T + 697 T^{2} + 28998 T^{3} + 1764324 T^{4} + 28998 p^{2} T^{5} + 697 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 7368 T^{3} - 239218 T^{4} - 7368 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 112 T + 3305 T^{2} + 121140 T^{3} - 10339324 T^{4} + 121140 p^{2} T^{5} + 3305 p^{4} T^{6} - 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 88 T + 2105 T^{2} + 102636 T^{3} - 8557612 T^{4} + 102636 p^{2} T^{5} + 2105 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 60 T + 2797 T^{2} - 95820 T^{3} + 1350408 T^{4} - 95820 p^{2} T^{5} + 2797 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 1068 T^{3} - 3628786 T^{4} - 1068 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 100 T + 7818 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 90 T + 2349 T^{2} - 371538 T^{3} - 33459160 T^{4} - 371538 p^{2} T^{5} + 2349 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 46 T - 5555 T^{2} - 10534 T^{3} + 37124764 T^{4} - 10534 p^{2} T^{5} - 5555 p^{4} T^{6} - 46 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 198 T + 20205 T^{2} - 1399650 T^{3} + 90955016 T^{4} - 1399650 p^{2} T^{5} + 20205 p^{4} T^{6} - 198 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 54 T + 15129 T^{2} + 764106 T^{3} + 105652784 T^{4} + 764106 p^{2} T^{5} + 15129 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 120 T + 7200 T^{2} - 693480 T^{3} + 66591182 T^{4} - 693480 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 56 T + 13074 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 280 T + 39200 T^{2} - 4657800 T^{3} + 458461934 T^{4} - 4657800 p^{2} T^{5} + 39200 p^{4} T^{6} - 280 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 68 T + 11357 T^{2} - 1221600 T^{3} + 93740204 T^{4} - 1221600 p^{2} T^{5} + 11357 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 72 T + 6921 T^{2} - 1202748 T^{3} - 83331436 T^{4} - 1202748 p^{2} T^{5} + 6921 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.919832432885376613802616169049, −8.695104325339647473779425327679, −8.264070918318842230587225942832, −7.87597239712892379012988064148, −7.65707279270413343191283396070, −7.47232348318021680821489719639, −7.15081611649142377461626311810, −6.90069088402802601694315569692, −6.54234339352749920688365281394, −6.21533133842338047652194584719, −6.20900159742455097363106123667, −5.96334850811203761983956811561, −5.17886074254414773046206470233, −5.09398229682498742957283014283, −4.94531465039497437591432900954, −4.38612824892621388495436189759, −4.28163216493309387437977657309, −4.01632330353417683397869376356, −3.92497645616364520440475359565, −2.47870062170676706511186329418, −2.40868335315299174258106568221, −2.31443061676969986809023519375, −2.09843505807579538922468031548, −0.802602498088065822560811429726, −0.18026749602526950940665950282,
0.18026749602526950940665950282, 0.802602498088065822560811429726, 2.09843505807579538922468031548, 2.31443061676969986809023519375, 2.40868335315299174258106568221, 2.47870062170676706511186329418, 3.92497645616364520440475359565, 4.01632330353417683397869376356, 4.28163216493309387437977657309, 4.38612824892621388495436189759, 4.94531465039497437591432900954, 5.09398229682498742957283014283, 5.17886074254414773046206470233, 5.96334850811203761983956811561, 6.20900159742455097363106123667, 6.21533133842338047652194584719, 6.54234339352749920688365281394, 6.90069088402802601694315569692, 7.15081611649142377461626311810, 7.47232348318021680821489719639, 7.65707279270413343191283396070, 7.87597239712892379012988064148, 8.264070918318842230587225942832, 8.695104325339647473779425327679, 8.919832432885376613802616169049
