| L(s) = 1 | + 2·2-s + 4-s + 2·5-s − 2·7-s − 2·8-s + 4·10-s + 2·11-s − 16·13-s − 4·14-s − 4·16-s − 10·17-s + 4·19-s + 2·20-s + 4·22-s + 2·23-s + 9·25-s − 32·26-s − 2·28-s − 4·31-s − 2·32-s − 20·34-s − 4·35-s − 8·37-s + 8·38-s − 4·40-s + 4·41-s + 2·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s − 0.707·8-s + 1.26·10-s + 0.603·11-s − 4.43·13-s − 1.06·14-s − 16-s − 2.42·17-s + 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.417·23-s + 9/5·25-s − 6.27·26-s − 0.377·28-s − 0.718·31-s − 0.353·32-s − 3.42·34-s − 0.676·35-s − 1.31·37-s + 1.29·38-s − 0.632·40-s + 0.624·41-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.336915992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.336915992\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T - p T^{2} + 2 T^{3} + 36 T^{4} + 2 p T^{5} - p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 10 T + 49 T^{2} + 10 p T^{3} + 36 p T^{4} + 10 p^{2} T^{5} + 49 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 18 T^{2} + 16 T^{3} + 491 T^{4} + 16 p T^{5} - 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 25 T^{2} + 34 T^{3} + 220 T^{4} + 34 p T^{5} - 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 272 T^{3} - 1421 T^{4} - 272 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 128 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 10 T - T^{2} + 70 T^{3} + 3292 T^{4} + 70 p T^{5} - p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 94 T^{2} + 896 T^{3} + 9867 T^{4} + 896 p T^{5} + 94 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T - 74 T^{2} + 112 T^{3} + 3675 T^{4} + 112 p T^{5} - 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 93 T^{2} - 42 T^{3} + 10724 T^{4} - 42 p T^{5} - 93 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10 T + 13 T^{2} + 470 T^{3} - 3620 T^{4} + 470 p T^{5} + 13 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 16 p T^{5} - 126 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 18 T + 103 T^{2} + 1134 T^{3} + 17004 T^{4} + 1134 p T^{5} + 103 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 14 T + 207 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 58 T^{2} + 448 T^{3} + 1267 T^{4} + 448 p T^{5} - 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 26 T + 355 T^{2} + 26 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.72154936986498187605117110998, −6.72077057347371523663387637081, −6.56569202647164996088199418031, −6.22618291882989683056977629327, −5.81244741522650276625272603062, −5.49057891625365421758971751259, −5.47412752164126205848966644415, −5.28979939127568708750041287234, −4.92873320033882545165258287102, −4.91916754104161788336795775564, −4.61817547246575413571789199545, −4.49925639639893120650247363208, −4.27276199434790416563689152513, −4.00041023137910432940469268598, −3.51848597231228586258581817514, −3.44163445302693279683894351679, −2.95874300717292081382872984800, −2.88633330558794700538444226758, −2.61357252844115598472692173870, −2.30229617999755516831390922854, −2.24456830159863625944498970796, −1.72812060966195884419057826533, −1.51528882943935289287348409276, −0.62620629884395417407779320090, −0.22659812491714184598963774322,
0.22659812491714184598963774322, 0.62620629884395417407779320090, 1.51528882943935289287348409276, 1.72812060966195884419057826533, 2.24456830159863625944498970796, 2.30229617999755516831390922854, 2.61357252844115598472692173870, 2.88633330558794700538444226758, 2.95874300717292081382872984800, 3.44163445302693279683894351679, 3.51848597231228586258581817514, 4.00041023137910432940469268598, 4.27276199434790416563689152513, 4.49925639639893120650247363208, 4.61817547246575413571789199545, 4.91916754104161788336795775564, 4.92873320033882545165258287102, 5.28979939127568708750041287234, 5.47412752164126205848966644415, 5.49057891625365421758971751259, 5.81244741522650276625272603062, 6.22618291882989683056977629327, 6.56569202647164996088199418031, 6.72077057347371523663387637081, 6.72154936986498187605117110998
