| L(s) = 1 | − 3·2-s − 3·3-s + 3·4-s + 9·6-s − 9·12-s − 3·16-s − 6·17-s + 6·23-s + 10·27-s + 15·29-s + 9·31-s + 6·32-s + 18·34-s + 12·41-s − 18·46-s + 6·47-s + 9·48-s − 18·49-s + 18·51-s − 6·53-s − 30·54-s − 45·58-s + 21·59-s + 9·61-s − 27·62-s − 8·64-s + 18·67-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3/2·4-s + 3.67·6-s − 2.59·12-s − 3/4·16-s − 1.45·17-s + 1.25·23-s + 1.92·27-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 + 1.29·48-s − 2.57·49-s + 2.52·51-s − 0.824·53-s − 4.08·54-s − 5.90·58-s + 2.73·59-s + 1.15·61-s − 3.42·62-s − 64-s + 2.19·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5233168780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5233168780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 + p T + p^{2} T^{2} + 17 T^{3} + p^{3} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 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
−6.82183911011556705288425729536, −6.57723296998921608867352947548, −6.48955054119287272296482768746, −6.35564674258792976471622172423, −6.10129187749498484145853534931, −5.70286344696898770416583917933, −5.33926835836793978679789879315, −5.22380728510170013619545349629, −5.05497788186320726354706561162, −4.96010219377992190620177576792, −4.59294923691610902385761941072, −4.22378456651606922424848232076, −4.15811814987130454005928909427, −3.61557607405237872523506004173, −3.59219918406655310276706902359, −2.90443495194438659223981052315, −2.71859496033189390527828707667, −2.58389911400335873020030407016, −2.31980406943432815231010789100, −1.97672829982684417082441784598, −1.31947679574013114427220349432, −1.08620044634322925894159128700, −0.60325252527394366487365240037, −0.59241006113173263225001089395, −0.46926915444142360084252697658,
0.46926915444142360084252697658, 0.59241006113173263225001089395, 0.60325252527394366487365240037, 1.08620044634322925894159128700, 1.31947679574013114427220349432, 1.97672829982684417082441784598, 2.31980406943432815231010789100, 2.58389911400335873020030407016, 2.71859496033189390527828707667, 2.90443495194438659223981052315, 3.59219918406655310276706902359, 3.61557607405237872523506004173, 4.15811814987130454005928909427, 4.22378456651606922424848232076, 4.59294923691610902385761941072, 4.96010219377992190620177576792, 5.05497788186320726354706561162, 5.22380728510170013619545349629, 5.33926835836793978679789879315, 5.70286344696898770416583917933, 6.10129187749498484145853534931, 6.35564674258792976471622172423, 6.48955054119287272296482768746, 6.57723296998921608867352947548, 6.82183911011556705288425729536
