| L(s) = 1 | − 16·7-s − 8·13-s − 32·19-s − 10·25-s + 8·31-s + 136·37-s − 80·43-s + 144·49-s + 40·61-s + 304·67-s + 152·73-s + 200·79-s + 128·91-s − 424·97-s − 112·103-s − 104·109-s + 88·121-s + 127-s + 131-s + 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.28·7-s − 0.615·13-s − 1.68·19-s − 2/5·25-s + 8/31·31-s + 3.67·37-s − 1.86·43-s + 2.93·49-s + 0.655·61-s + 4.53·67-s + 2.08·73-s + 2.53·79-s + 1.40·91-s − 4.37·97-s − 1.08·103-s − 0.954·109-s + 8/11·121-s + 0.00787·127-s + 0.00763·131-s + 3.84·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7531124876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7531124876\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 8 T + 24 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 p T^{2} + 18258 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 252 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 16 T + 426 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1612 T^{2} + 1157478 T^{4} - 1612 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 700 T^{2} + 707622 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 1566 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 68 T + 3084 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 640 T^{2} + 3887682 T^{4} - 640 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 40 T + 3738 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 340 T^{2} - 372378 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1876 T^{2} - 4075194 T^{4} - 1876 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 280 T^{2} - 9454638 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 4302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 76 T + 8862 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 100 T + 11742 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 21652 T^{2} + 211289478 T^{4} - 21652 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 212 T + 29694 T^{2} + 212 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.21881703075578863977763578287, −5.84244726416050522283063949889, −5.68204346820005332631690490509, −5.31458991174551035590850879045, −5.25080577518498545709925877548, −5.20690837318545624164673372360, −4.71662452877760202553494275009, −4.62020967541014593362133648943, −4.17974991680375483144716163003, −4.17916768644048147655878025408, −3.86216652590826416103297660016, −3.79802390209566812433652428655, −3.45366884666049976908480489175, −3.40813934341592492468609471910, −2.91383745913271619718211748796, −2.67861688137078239359607768648, −2.52074632149890820205208226825, −2.51666465656612974725774855539, −2.05427626253472925262526158507, −1.93544225375136720590782335511, −1.47579951102444598491255350780, −0.882258569866188961204681124014, −0.866539520986737301996674551145, −0.49749027980153699482417549770, −0.13796071161197140231592230668,
0.13796071161197140231592230668, 0.49749027980153699482417549770, 0.866539520986737301996674551145, 0.882258569866188961204681124014, 1.47579951102444598491255350780, 1.93544225375136720590782335511, 2.05427626253472925262526158507, 2.51666465656612974725774855539, 2.52074632149890820205208226825, 2.67861688137078239359607768648, 2.91383745913271619718211748796, 3.40813934341592492468609471910, 3.45366884666049976908480489175, 3.79802390209566812433652428655, 3.86216652590826416103297660016, 4.17916768644048147655878025408, 4.17974991680375483144716163003, 4.62020967541014593362133648943, 4.71662452877760202553494275009, 5.20690837318545624164673372360, 5.25080577518498545709925877548, 5.31458991174551035590850879045, 5.68204346820005332631690490509, 5.84244726416050522283063949889, 6.21881703075578863977763578287
