| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s − 9·6-s − 4·7-s + 10·8-s + 9-s − 11-s − 18·12-s − 12·13-s − 12·14-s + 15·16-s + 17-s + 3·18-s + 3·19-s + 12·21-s − 3·22-s − 8·23-s − 30·24-s − 36·26-s + 8·27-s − 24·28-s − 2·29-s − 6·31-s + 21·32-s + 3·33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 3.67·6-s − 1.51·7-s + 3.53·8-s + 1/3·9-s − 0.301·11-s − 5.19·12-s − 3.32·13-s − 3.20·14-s + 15/4·16-s + 0.242·17-s + 0.707·18-s + 0.688·19-s + 2.61·21-s − 0.639·22-s − 1.66·23-s − 6.12·24-s − 7.06·26-s + 1.53·27-s − 4.53·28-s − 0.371·29-s − 1.07·31-s + 3.71·32-s + 0.522·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 8 T^{2} + 13 T^{3} + 8 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 13 T^{2} + 22 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + T + 10 T^{2} - 3 T^{3} + 10 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 12 T + 83 T^{2} + 358 T^{3} + 83 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 30 T^{2} - 63 T^{3} + 30 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 32 T^{2} - 35 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 336 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 318 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 21 T + 254 T^{2} + 1937 T^{3} + 254 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + T^{2} - 388 T^{3} + p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 93 T^{2} + 312 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 59 T^{2} + 4 T^{3} + 59 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 49 T^{2} + 132 T^{3} + 49 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 139 T^{2} + 686 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 13 T + 192 T^{2} + 25 p T^{3} + 192 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 169 T^{2} + 846 T^{3} + 169 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 31 T + 534 T^{2} + 5575 T^{3} + 534 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 22 T + 377 T^{2} - 3708 T^{3} + 377 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 7 T + 22 T^{2} - 657 T^{3} + 22 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 240 T^{2} + 507 T^{3} + 240 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 287 T^{2} + 1124 T^{3} + 287 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.494031857496862585393605755609, −8.166922527811166607370359984698, −7.956457332172075125173072434867, −7.41548929314275333717262350046, −7.26658472144242846198002911503, −7.17871171923660310165916318813, −6.96556838448942531042278721785, −6.40785168792858709843699001390, −6.38647032776418567834371524018, −6.14645281315181307945576453784, −5.73135121159042644826580624446, −5.54105147213579645505073727464, −5.41035212151330347280175613963, −4.95422430234376365463207186813, −4.84772763250043229420889753281, −4.81663355257801189177141218422, −4.22090296842033137633880042530, −3.92460254527544979373050496392, −3.44971080724522410560992657218, −3.12046170213099853606966350508, −3.04175196976323395826458513540, −2.73270499268890823377421484147, −2.18772308432101248786939804207, −1.83267338175141695976612589082, −1.48290435010814020346951263517, 0, 0, 0,
1.48290435010814020346951263517, 1.83267338175141695976612589082, 2.18772308432101248786939804207, 2.73270499268890823377421484147, 3.04175196976323395826458513540, 3.12046170213099853606966350508, 3.44971080724522410560992657218, 3.92460254527544979373050496392, 4.22090296842033137633880042530, 4.81663355257801189177141218422, 4.84772763250043229420889753281, 4.95422430234376365463207186813, 5.41035212151330347280175613963, 5.54105147213579645505073727464, 5.73135121159042644826580624446, 6.14645281315181307945576453784, 6.38647032776418567834371524018, 6.40785168792858709843699001390, 6.96556838448942531042278721785, 7.17871171923660310165916318813, 7.26658472144242846198002911503, 7.41548929314275333717262350046, 7.956457332172075125173072434867, 8.166922527811166607370359984698, 8.494031857496862585393605755609
