| L(s) = 1 | − 3·2-s + 6·4-s − 3·7-s − 10·8-s + 6·13-s + 9·14-s + 15·16-s − 6·17-s − 6·23-s − 6·25-s − 18·26-s − 18·28-s − 6·29-s + 9·31-s − 21·32-s + 18·34-s + 9·37-s − 9·41-s + 15·43-s + 18·46-s − 3·47-s − 12·49-s + 18·50-s + 36·52-s + 3·53-s + 30·56-s + 18·58-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 1.13·7-s − 3.53·8-s + 1.66·13-s + 2.40·14-s + 15/4·16-s − 1.45·17-s − 1.25·23-s − 6/5·25-s − 3.53·26-s − 3.40·28-s − 1.11·29-s + 1.61·31-s − 3.71·32-s + 3.08·34-s + 1.47·37-s − 1.40·41-s + 2.28·43-s + 2.65·46-s − 0.437·47-s − 1.71·49-s + 2.54·50-s + 4.99·52-s + 0.412·53-s + 4.00·56-s + 2.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 6 T^{2} - 9 T^{3} + 6 p T^{4} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 41 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 30 T^{2} + T^{3} + 30 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 6 T + 48 T^{2} - 159 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 11 p T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 7 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 135 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 531 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_6$ | \( 1 - 9 T + 54 T^{2} - 305 T^{3} + 54 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 111 T^{2} + 629 T^{3} + 111 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 15 T + 183 T^{2} - 1347 T^{3} + 183 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 117 T^{2} + 229 T^{3} + 117 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 3 T + 6 T^{2} - 11 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 12 T + 144 T^{2} - 1399 T^{3} + 144 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 33 T + 543 T^{2} + 5323 T^{3} + 543 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 210 T^{2} - 1605 T^{3} + 210 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 15 T + 159 T^{2} + 1681 T^{3} + 159 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 475 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 15 T + 87 T^{2} - 245 T^{3} + 87 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 27 T + 471 T^{2} + 4985 T^{3} + 471 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 15 T + 231 T^{2} - 2563 T^{3} + 231 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 12 T + 336 T^{2} + 2381 T^{3} + 336 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.65574445211818549620194626210, −7.31133136156651192089727612828, −7.08259794853776733304369705162, −6.69943888902637032760917298960, −6.45283837792039011423210069277, −6.26258968983960163447489869206, −6.19183367954046071205937598891, −6.00975614005208078233114001212, −5.89022044829534298197370366106, −5.55675555160251693772086658606, −4.87595849497856485869959294914, −4.87485374110668141927162985839, −4.58490468497241673907751222306, −3.94037631869557974020537035795, −3.93464010532017230112595200467, −3.81602628428839575109205854166, −3.21242043133111311202655672689, −3.20906583157573215964850325745, −2.81695505660309730131604407174, −2.25893988121616764566448755488, −2.20658112104966058744225493887, −2.13982565431257099513408520199, −1.34977155157565873006680932687, −1.19079801117921549999386540851, −1.08508330011487458343229041718, 0, 0, 0,
1.08508330011487458343229041718, 1.19079801117921549999386540851, 1.34977155157565873006680932687, 2.13982565431257099513408520199, 2.20658112104966058744225493887, 2.25893988121616764566448755488, 2.81695505660309730131604407174, 3.20906583157573215964850325745, 3.21242043133111311202655672689, 3.81602628428839575109205854166, 3.93464010532017230112595200467, 3.94037631869557974020537035795, 4.58490468497241673907751222306, 4.87485374110668141927162985839, 4.87595849497856485869959294914, 5.55675555160251693772086658606, 5.89022044829534298197370366106, 6.00975614005208078233114001212, 6.19183367954046071205937598891, 6.26258968983960163447489869206, 6.45283837792039011423210069277, 6.69943888902637032760917298960, 7.08259794853776733304369705162, 7.31133136156651192089727612828, 7.65574445211818549620194626210
