| L(s) = 1 | − 3·2-s + 6·4-s − 5·5-s + 7-s − 10·8-s + 15·10-s + 3·11-s + 5·13-s − 3·14-s + 15·16-s − 4·17-s − 3·19-s − 30·20-s − 9·22-s − 2·23-s + 7·25-s − 15·26-s + 6·28-s − 7·29-s − 3·31-s − 21·32-s + 12·34-s − 5·35-s − 14·37-s + 9·38-s + 50·40-s + 3·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 2.23·5-s + 0.377·7-s − 3.53·8-s + 4.74·10-s + 0.904·11-s + 1.38·13-s − 0.801·14-s + 15/4·16-s − 0.970·17-s − 0.688·19-s − 6.70·20-s − 1.91·22-s − 0.417·23-s + 7/5·25-s − 2.94·26-s + 1.13·28-s − 1.29·29-s − 0.538·31-s − 3.71·32-s + 2.05·34-s − 0.845·35-s − 2.30·37-s + 1.45·38-s + 7.90·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + p T + 18 T^{2} + 48 T^{3} + 18 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 40 T^{2} - 116 T^{3} + 40 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 27 T^{2} + 48 T^{3} + 27 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 41 T^{2} + 124 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 68 T^{2} + 320 T^{3} + 68 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 68 T^{2} + 110 T^{3} + 68 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 14 T + 127 T^{2} + 908 T^{3} + 127 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 3 T + 98 T^{2} - 268 T^{3} + 98 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 62 T^{2} + 162 T^{3} + 62 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 89 T^{2} - 128 T^{3} + 89 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 16 T + 223 T^{2} - 1752 T^{3} + 223 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 17 T^{2} + 376 T^{3} + 17 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + 194 T^{2} - 138 T^{3} + 194 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 3 T + 182 T^{2} - 454 T^{3} + 182 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T + 83 T^{2} - 528 T^{3} + 83 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 121 T^{2} - 668 T^{3} + 121 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 7 T + 106 T^{2} + 54 T^{3} + 106 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 16 T - 13 T^{2} - 1576 T^{3} - 13 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 307 T^{2} + 2588 T^{3} + 307 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.192929362828945632658837673618, −7.57359412683177569091710650648, −7.44410264236533829424795803660, −7.40544782079716733687653102695, −7.00622091272332086964681069900, −6.92944690877421475483224234596, −6.57748010026919893675054138239, −6.26425868437162378201018781219, −6.20354648492667018943797692968, −5.82803228186317897629811976676, −5.26708647092710852141868282063, −5.25640374435267706145709799033, −5.00717385766238475280726972118, −4.17708255346531472322744190190, −4.15265529954075732588312975502, −4.03841923813607529975425364823, −3.57613397420981785457101836346, −3.51779183819064320138152705499, −3.36205253868055070694629869591, −2.51936507360246122444645169819, −2.35041655757658039750242182638, −2.14977050317208082753888242983, −1.45990279284914637066180626403, −1.35540547323113083649514877487, −1.07057201446505722551380224014, 0, 0, 0,
1.07057201446505722551380224014, 1.35540547323113083649514877487, 1.45990279284914637066180626403, 2.14977050317208082753888242983, 2.35041655757658039750242182638, 2.51936507360246122444645169819, 3.36205253868055070694629869591, 3.51779183819064320138152705499, 3.57613397420981785457101836346, 4.03841923813607529975425364823, 4.15265529954075732588312975502, 4.17708255346531472322744190190, 5.00717385766238475280726972118, 5.25640374435267706145709799033, 5.26708647092710852141868282063, 5.82803228186317897629811976676, 6.20354648492667018943797692968, 6.26425868437162378201018781219, 6.57748010026919893675054138239, 6.92944690877421475483224234596, 7.00622091272332086964681069900, 7.40544782079716733687653102695, 7.44410264236533829424795803660, 7.57359412683177569091710650648, 8.192929362828945632658837673618
