| L(s) = 1 | − 4·3-s − 6·9-s + 48·13-s − 56·19-s − 4·25-s + 76·27-s + 24·31-s + 120·37-s − 192·39-s − 120·43-s − 20·49-s + 224·57-s − 48·61-s + 136·67-s − 104·73-s + 16·75-s − 192·79-s − 109·81-s − 96·93-s − 360·97-s + 168·103-s + 128·109-s − 480·111-s − 288·117-s − 22·121-s + 127-s + 480·129-s + ⋯ |
| L(s) = 1 | − 4/3·3-s − 2/3·9-s + 3.69·13-s − 2.94·19-s − 0.159·25-s + 2.81·27-s + 0.774·31-s + 3.24·37-s − 4.92·39-s − 2.79·43-s − 0.408·49-s + 3.92·57-s − 0.786·61-s + 2.02·67-s − 1.42·73-s + 0.213·75-s − 2.43·79-s − 1.34·81-s − 1.03·93-s − 3.71·97-s + 1.63·103-s + 1.17·109-s − 4.32·111-s − 2.46·117-s − 0.181·121-s + 0.00787·127-s + 3.72·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9677816648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9677816648\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 154 T^{4} + 4 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 24 T + 394 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 548 T^{2} + 152006 T^{4} - 548 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 28 T + 566 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1180 T^{2} + 793734 T^{4} - 1180 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2948 T^{2} + 3564710 T^{4} - 2948 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 1606 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 60 T + 3286 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 3236 T^{2} + 7165574 T^{4} - 3236 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 60 T + 4246 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 7964 T^{2} + 25546694 T^{4} - 7964 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6524 T^{2} + 24696806 T^{4} - 6524 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1540 T^{2} - 4368666 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 24 T + 7498 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 68 T + 4502 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8732 T^{2} + 61537286 T^{4} - 8732 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 52 T + 2534 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 96 T + 14698 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6652 T^{2} + 94061606 T^{4} + 6652 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 29732 T^{2} + 345919238 T^{4} - 29732 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 180 T + 25510 T^{2} + 180 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−9.577990669632024711119150319558, −9.314897175201610425668146588946, −8.691942293754224939086056408306, −8.515631402310660871799254634134, −8.409367642329261697807057404175, −8.318785141006575132198691118353, −8.166025228394766077126418199134, −7.56246846318674985110626659429, −6.92152331138720457126023628021, −6.62073772941342379783827933792, −6.58746093998387031164345176793, −6.14367602458893349767216016208, −5.95989970095986457000561774967, −5.86172324499724082875395473454, −5.59786928759725336060040295557, −4.89118291830986800707641256653, −4.64035691952586383280035149383, −4.12597517193238478663924325190, −4.11139251941347730670543369472, −3.43004255918660738661113053838, −3.04744421348196799514269874466, −2.60500773002384297956998016340, −1.80845641971328716610752250374, −1.23295726433643618389422352550, −0.46764483837531875041946054064,
0.46764483837531875041946054064, 1.23295726433643618389422352550, 1.80845641971328716610752250374, 2.60500773002384297956998016340, 3.04744421348196799514269874466, 3.43004255918660738661113053838, 4.11139251941347730670543369472, 4.12597517193238478663924325190, 4.64035691952586383280035149383, 4.89118291830986800707641256653, 5.59786928759725336060040295557, 5.86172324499724082875395473454, 5.95989970095986457000561774967, 6.14367602458893349767216016208, 6.58746093998387031164345176793, 6.62073772941342379783827933792, 6.92152331138720457126023628021, 7.56246846318674985110626659429, 8.166025228394766077126418199134, 8.318785141006575132198691118353, 8.409367642329261697807057404175, 8.515631402310660871799254634134, 8.691942293754224939086056408306, 9.314897175201610425668146588946, 9.577990669632024711119150319558
