| L(s) = 1 | − 3·7-s − 6·9-s + 3·11-s − 6·23-s − 12·25-s + 27-s + 12·29-s − 3·31-s − 6·37-s − 12·41-s − 6·43-s − 6·47-s − 3·49-s − 12·53-s − 6·59-s − 12·61-s + 18·63-s + 18·67-s + 18·71-s − 24·73-s − 9·77-s + 24·79-s + 18·81-s − 15·83-s − 18·99-s − 12·101-s + 3·103-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 2·9-s + 0.904·11-s − 1.25·23-s − 2.39·25-s + 0.192·27-s + 2.22·29-s − 0.538·31-s − 0.986·37-s − 1.87·41-s − 0.914·43-s − 0.875·47-s − 3/7·49-s − 1.64·53-s − 0.781·59-s − 1.53·61-s + 2.26·63-s + 2.19·67-s + 2.13·71-s − 2.80·73-s − 1.02·77-s + 2.70·79-s + 2·81-s − 1.64·83-s − 1.80·99-s − 1.19·101-s + 0.295·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} - T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 12 T^{2} + T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 12 T^{2} + 39 T^{3} + 12 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 24 T^{2} - 63 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 36 T^{2} + T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 48 T^{2} + T^{3} + 48 p T^{4} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 225 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 623 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 3 T + 60 T^{2} + 79 T^{3} + 60 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 292 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $A_4\times C_2$ | \( 1 + 6 T + 120 T^{2} + 499 T^{3} + 120 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 6 T + 114 T^{2} + 405 T^{3} + 114 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 569 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 114 T^{2} + 691 T^{3} + 114 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 12 T + 84 T^{2} + 257 T^{3} + 84 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 18 T + 282 T^{2} - 2439 T^{3} + 282 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 18 T + 204 T^{2} - 1593 T^{3} + 204 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 24 T + 264 T^{2} + 2157 T^{3} + 264 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 24 T + 402 T^{2} - 4061 T^{3} + 402 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 15 T + 72 T^{2} - 13 T^{3} + 72 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 240 T^{2} - 27 T^{3} + 240 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 216 T^{2} + 125 T^{3} + 216 p T^{4} + p^{3} T^{6} \) |
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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
−8.301884016490335083007172149051, −7.79241321178618898777449452961, −7.72754730808552913716945316699, −7.70168836227077978385743036183, −6.84091179673199380699472615617, −6.83121654320663435271951929151, −6.50261980293782961947609480047, −6.27907970531489713457300804665, −6.27404893436909617599910166066, −6.06113329679422389654284229946, −5.46312714301199830960637662783, −5.34313416931971496385776554454, −5.16104252602411316753841459678, −4.71440322276550300204380181688, −4.62018398821716743298830091210, −3.87604199104350918080722660271, −3.67875943273078127878859484532, −3.66639876497587420387677613770, −3.46026821692991231785918825610, −2.72393667414996099819406091508, −2.66599583693409744648047092664, −2.59370886379618239346426044468, −1.77137934174660693392925988658, −1.57483238642487701502563308613, −1.23893210827327695588887692437, 0, 0, 0,
1.23893210827327695588887692437, 1.57483238642487701502563308613, 1.77137934174660693392925988658, 2.59370886379618239346426044468, 2.66599583693409744648047092664, 2.72393667414996099819406091508, 3.46026821692991231785918825610, 3.66639876497587420387677613770, 3.67875943273078127878859484532, 3.87604199104350918080722660271, 4.62018398821716743298830091210, 4.71440322276550300204380181688, 5.16104252602411316753841459678, 5.34313416931971496385776554454, 5.46312714301199830960637662783, 6.06113329679422389654284229946, 6.27404893436909617599910166066, 6.27907970531489713457300804665, 6.50261980293782961947609480047, 6.83121654320663435271951929151, 6.84091179673199380699472615617, 7.70168836227077978385743036183, 7.72754730808552913716945316699, 7.79241321178618898777449452961, 8.301884016490335083007172149051
