| L(s) = 1 | − 6·9-s − 3·11-s − 3·13-s + 6·17-s + 3·19-s − 6·23-s + 27-s − 3·29-s − 15·31-s + 9·37-s − 9·41-s − 3·43-s − 3·47-s − 12·49-s − 3·53-s + 6·59-s − 9·61-s + 3·67-s − 9·71-s − 9·73-s − 12·79-s + 18·81-s − 9·83-s + 6·89-s + 21·97-s + 18·99-s − 12·101-s + ⋯ |
| L(s) = 1 | − 2·9-s − 0.904·11-s − 0.832·13-s + 1.45·17-s + 0.688·19-s − 1.25·23-s + 0.192·27-s − 0.557·29-s − 2.69·31-s + 1.47·37-s − 1.40·41-s − 0.457·43-s − 0.437·47-s − 1.71·49-s − 0.412·53-s + 0.781·59-s − 1.15·61-s + 0.366·67-s − 1.06·71-s − 1.05·73-s − 1.35·79-s + 2·81-s − 0.987·83-s + 0.635·89-s + 2.13·97-s + 1.80·99-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} - T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 12 T^{2} - 9 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 27 T^{2} + 49 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 95 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 205 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 187 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 225 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 15 T + 159 T^{2} + 1019 T^{3} + 159 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 9 T + 81 T^{2} - 17 p T^{3} + 81 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 111 T^{2} + 629 T^{3} + 111 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 120 T^{2} + 239 T^{3} + 120 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 299 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 3 T + 114 T^{2} + 207 T^{3} + 114 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 412 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 865 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 3 T + 195 T^{2} - 385 T^{3} + 195 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 192 T^{2} + 1097 T^{3} + 192 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 9 T + 237 T^{2} + 1323 T^{3} + 237 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 12 T + 258 T^{2} + 1825 T^{3} + 258 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 9 T + 147 T^{2} + 1205 T^{3} + 147 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 6 T + 258 T^{2} - 1017 T^{3} + 258 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 21 T + 381 T^{2} - 4181 T^{3} + 381 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.901326229438871421149250964558, −7.64448322289228217288707162514, −7.53192541433283629983734259510, −7.38200234923026199192949148490, −6.79490506050379358829243719492, −6.69019412647893556108504282210, −6.44088983363705150530449255541, −5.90712600264842395682485759048, −5.86000348235428131809255678835, −5.69242295957636722024627430370, −5.28366232558065771270557306381, −5.18711814676500643671076240438, −5.17399417349515170705472935268, −4.47236605857533314927640189871, −4.46366786291100142846361906823, −3.94763254075985040351879470846, −3.44220821863665790200337104837, −3.43917364238334654793305555692, −3.26686989886481837304472025698, −2.72581053548305303629416717507, −2.55472612199139275649697105099, −2.38042796625564766531525606693, −1.72958467717516518021661859720, −1.51832112309594158527862142006, −1.13527495865770761705306152221, 0, 0, 0,
1.13527495865770761705306152221, 1.51832112309594158527862142006, 1.72958467717516518021661859720, 2.38042796625564766531525606693, 2.55472612199139275649697105099, 2.72581053548305303629416717507, 3.26686989886481837304472025698, 3.43917364238334654793305555692, 3.44220821863665790200337104837, 3.94763254075985040351879470846, 4.46366786291100142846361906823, 4.47236605857533314927640189871, 5.17399417349515170705472935268, 5.18711814676500643671076240438, 5.28366232558065771270557306381, 5.69242295957636722024627430370, 5.86000348235428131809255678835, 5.90712600264842395682485759048, 6.44088983363705150530449255541, 6.69019412647893556108504282210, 6.79490506050379358829243719492, 7.38200234923026199192949148490, 7.53192541433283629983734259510, 7.64448322289228217288707162514, 7.901326229438871421149250964558
