| L(s) = 1 | − 3·2-s + 3·4-s + 3·5-s − 9·10-s − 3·16-s − 6·17-s + 9·20-s + 6·23-s − 6·25-s − 15·29-s + 9·31-s + 6·32-s + 18·34-s − 12·41-s − 18·46-s + 6·47-s − 18·49-s + 18·50-s − 6·53-s + 45·58-s − 21·59-s + 9·61-s − 27·62-s − 8·64-s − 18·67-s − 18·68-s − 30·71-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 1.34·5-s − 2.84·10-s − 3/4·16-s − 1.45·17-s + 2.01·20-s + 1.25·23-s − 6/5·25-s − 2.78·29-s + 1.61·31-s + 1.06·32-s + 3.08·34-s − 1.87·41-s − 2.65·46-s + 0.875·47-s − 2.57·49-s + 2.54·50-s − 0.824·53-s + 5.90·58-s − 2.73·59-s + 1.15·61-s − 3.42·62-s − 64-s − 2.19·67-s − 2.18·68-s − 3.56·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 27 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 18 T^{2} + T^{3} + 18 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 24 T^{2} + 9 T^{3} + 24 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 18 T^{2} - 37 T^{3} + 18 p T^{4} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 252 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 15 T + 159 T^{2} + 981 T^{3} + 159 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 9 T + 99 T^{2} - 505 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 90 T^{2} + 17 T^{3} + 90 p T^{4} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 873 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 72 T^{2} + 163 T^{3} + 72 p T^{4} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 567 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 585 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 21 T + 312 T^{2} + 2745 T^{3} + 312 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 9 T + 162 T^{2} - 917 T^{3} + 162 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 18 T + 225 T^{2} + 1988 T^{3} + 225 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 30 T + 501 T^{2} + 5148 T^{3} + 501 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 171 T^{2} + 64 T^{3} + 171 p T^{4} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 9 T + 135 T^{2} - 613 T^{3} + 135 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 60 T^{2} - 459 T^{3} + 60 p T^{4} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 15 T + 321 T^{2} + 2727 T^{3} + 321 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 15 T + 330 T^{2} + 2783 T^{3} + 330 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.315840332685928637162943891240, −7.72030734271086245669972859197, −7.64306275960126340385388965492, −7.51647941811671633759943683184, −7.12883891160506448375809713421, −6.96009124499980283078293197402, −6.44192460960282242635161023644, −6.28196186184383490713852097857, −6.16350657416401424004217405162, −6.02223617538331563361682438815, −5.41381228025841099546152283420, −5.32702568162240647333683330524, −5.11035727323969968715847445350, −4.47979811148684248884135718600, −4.43304594858105452066563366195, −4.37980595115564184894805550310, −3.59466247994674825738193120485, −3.50951391285361150307046508669, −3.02364341133847395127289265924, −2.78345063772544066344642407782, −2.28843287625433641569469647957, −2.07119106061965853642348112479, −1.60491843963965330045928705643, −1.41622820564335959410394786195, −1.21908970593595420693185808000, 0, 0, 0,
1.21908970593595420693185808000, 1.41622820564335959410394786195, 1.60491843963965330045928705643, 2.07119106061965853642348112479, 2.28843287625433641569469647957, 2.78345063772544066344642407782, 3.02364341133847395127289265924, 3.50951391285361150307046508669, 3.59466247994674825738193120485, 4.37980595115564184894805550310, 4.43304594858105452066563366195, 4.47979811148684248884135718600, 5.11035727323969968715847445350, 5.32702568162240647333683330524, 5.41381228025841099546152283420, 6.02223617538331563361682438815, 6.16350657416401424004217405162, 6.28196186184383490713852097857, 6.44192460960282242635161023644, 6.96009124499980283078293197402, 7.12883891160506448375809713421, 7.51647941811671633759943683184, 7.64306275960126340385388965492, 7.72030734271086245669972859197, 8.315840332685928637162943891240
