| L(s) = 1 | + 2·2-s − 4-s − 3·5-s − 5·7-s − 5·8-s − 6·10-s + 2·11-s − 5·13-s − 10·14-s − 16-s + 5·17-s − 14·19-s + 3·20-s + 4·22-s + 4·23-s + 6·25-s − 10·26-s + 5·28-s + 15·29-s − 6·31-s + 4·32-s + 10·34-s + 15·35-s + 5·37-s − 28·38-s + 15·40-s + 17·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 1.34·5-s − 1.88·7-s − 1.76·8-s − 1.89·10-s + 0.603·11-s − 1.38·13-s − 2.67·14-s − 1/4·16-s + 1.21·17-s − 3.21·19-s + 0.670·20-s + 0.852·22-s + 0.834·23-s + 6/5·25-s − 1.96·26-s + 0.944·28-s + 2.78·29-s − 1.07·31-s + 0.707·32-s + 1.71·34-s + 2.53·35-s + 0.821·37-s − 4.54·38-s + 2.37·40-s + 2.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 89^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 89^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 89 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $A_4\times C_2$ | \( 1 - p T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 5 T + 27 T^{2} + 71 T^{3} + 27 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 2 T + 25 T^{2} - 36 T^{3} + 25 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 5 T + 45 T^{2} + 131 T^{3} + 45 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 5 T + 43 T^{2} - 171 T^{3} + 43 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 14 T + 113 T^{2} + 588 T^{3} + 113 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 4 T + 65 T^{2} - 176 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 15 T + 141 T^{2} - 883 T^{3} + 141 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 380 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 5 T + 103 T^{2} - 371 T^{3} + 103 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 17 T + 189 T^{2} - 1437 T^{3} + 189 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 11 T + 167 T^{2} + 987 T^{3} + 167 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 11 T + 151 T^{2} + 1005 T^{3} + 151 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 11 T + 127 T^{2} - 1095 T^{3} + 127 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 3 T + 5 T^{2} - 695 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 16 T + 259 T^{2} + 2056 T^{3} + 259 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 185 T^{2} - 700 T^{3} + 185 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 10 T + 125 T^{2} + 644 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 16 T + 239 T^{2} + 2328 T^{3} + 239 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 7 T + 41 T^{2} + 525 T^{3} + 41 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 14 T + 53 T^{2} + 28 T^{3} + 53 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 4 T + 147 T^{2} - 712 T^{3} + 147 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.88595200541469378691694752218, −7.58251871218208636548757950359, −7.27257715417755632597527213221, −7.00937077609354471651279634298, −6.72216429381638992258726676342, −6.59882491676749603789948137676, −6.40884129488668031459091922702, −5.91228167203727121124133962170, −5.84282445159269394357551924780, −5.80090513618947926729296587632, −5.00191437530549149505854175546, −4.93122562434269208015111590745, −4.75343147051745797926058696862, −4.47101209065029654878708736420, −4.27709725495812923029753484712, −4.18060131368958357040778103409, −3.67521162205355630118483023776, −3.51706888271231032431076576640, −3.36249613406631931611864051014, −2.80310273691958775510524525552, −2.75625680868281670302821181155, −2.59484714148954947059025313772, −1.88522264032436743898283087339, −1.28278489254235195100501665064, −1.00143698708267362818515375017, 0, 0, 0,
1.00143698708267362818515375017, 1.28278489254235195100501665064, 1.88522264032436743898283087339, 2.59484714148954947059025313772, 2.75625680868281670302821181155, 2.80310273691958775510524525552, 3.36249613406631931611864051014, 3.51706888271231032431076576640, 3.67521162205355630118483023776, 4.18060131368958357040778103409, 4.27709725495812923029753484712, 4.47101209065029654878708736420, 4.75343147051745797926058696862, 4.93122562434269208015111590745, 5.00191437530549149505854175546, 5.80090513618947926729296587632, 5.84282445159269394357551924780, 5.91228167203727121124133962170, 6.40884129488668031459091922702, 6.59882491676749603789948137676, 6.72216429381638992258726676342, 7.00937077609354471651279634298, 7.27257715417755632597527213221, 7.58251871218208636548757950359, 7.88595200541469378691694752218
