| L(s) = 1 | + 3·3-s − 6·5-s + 2·7-s + 6·9-s + 5·11-s − 18·15-s − 17-s − 7·19-s + 6·21-s + 16·25-s + 10·27-s − 2·29-s − 16·31-s + 15·33-s − 12·35-s − 22·37-s − 11·41-s + 15·43-s − 36·45-s + 7·47-s − 16·49-s − 3·51-s − 17·53-s − 30·55-s − 21·57-s − 6·59-s − 13·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 2.68·5-s + 0.755·7-s + 2·9-s + 1.50·11-s − 4.64·15-s − 0.242·17-s − 1.60·19-s + 1.30·21-s + 16/5·25-s + 1.92·27-s − 0.371·29-s − 2.87·31-s + 2.61·33-s − 2.02·35-s − 3.61·37-s − 1.71·41-s + 2.28·43-s − 5.36·45-s + 1.02·47-s − 2.28·49-s − 0.420·51-s − 2.33·53-s − 4.04·55-s − 2.78·57-s − 0.781·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 47 T^{3} + 4 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 2 T + 20 T^{2} - 27 T^{3} + 20 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 25 T^{2} - 69 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 7 T + 71 T^{2} + 273 T^{3} + 71 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 20 T^{2} - 91 T^{3} + 20 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 2 T + 72 T^{2} + 3 p T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 16 T + 134 T^{2} + 795 T^{3} + 134 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 22 T + 270 T^{2} + 2005 T^{3} + 270 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 11 T + 147 T^{2} + 873 T^{3} + 147 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 15 T + 176 T^{2} - 1331 T^{3} + 176 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 7 T + 155 T^{2} - 665 T^{3} + 155 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 17 T + 225 T^{2} + 1761 T^{3} + 225 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 161 T^{2} + 604 T^{3} + 161 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 11 T + 155 T^{2} - 1515 T^{3} + 155 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 122 T^{2} + 203 T^{3} + 122 p T^{4} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 6 T + 84 T^{2} - 47 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 219 T^{2} + 447 T^{3} + 219 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 12 T + 290 T^{2} - 2035 T^{3} + 290 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + T + 167 T^{2} + 65 T^{3} + 167 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 5 T + 10 T^{2} + 667 T^{3} + 10 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.46229967118417721759462569919, −7.03516817429594161215496031163, −6.97311434429082738836803092853, −6.85468850858029517633764372090, −6.30993747590563584644678823057, −6.25993099461592430194894140017, −6.09327287000248848040981267008, −5.51142072048776315736012774662, −5.18681186638454958227872970946, −5.03080508761883406185666462056, −4.71932765450182150952357328142, −4.44506537127265447257922927041, −4.42183188683509781022088495659, −3.98182263713176879400324312065, −3.71916259593973279325324547468, −3.65182188159323847733357319478, −3.53470982871237471414985492861, −3.26644205225937058848236915117, −3.10203284691290004949931396622, −2.49872049755187290361324842361, −2.10585158819220440917323077734, −1.98238708784353215785109961113, −1.67710493702141212673445693970, −1.36910959320106216313627902435, −1.10383014347914377542057420380, 0, 0, 0,
1.10383014347914377542057420380, 1.36910959320106216313627902435, 1.67710493702141212673445693970, 1.98238708784353215785109961113, 2.10585158819220440917323077734, 2.49872049755187290361324842361, 3.10203284691290004949931396622, 3.26644205225937058848236915117, 3.53470982871237471414985492861, 3.65182188159323847733357319478, 3.71916259593973279325324547468, 3.98182263713176879400324312065, 4.42183188683509781022088495659, 4.44506537127265447257922927041, 4.71932765450182150952357328142, 5.03080508761883406185666462056, 5.18681186638454958227872970946, 5.51142072048776315736012774662, 6.09327287000248848040981267008, 6.25993099461592430194894140017, 6.30993747590563584644678823057, 6.85468850858029517633764372090, 6.97311434429082738836803092853, 7.03516817429594161215496031163, 7.46229967118417721759462569919
