| L(s) = 1 | − 10·3-s + 20·5-s − 8·7-s + 56·9-s + 86·11-s − 124·13-s − 200·15-s + 14·17-s + 88·19-s + 80·21-s − 68·23-s + 68·25-s − 248·27-s − 87·29-s − 326·31-s − 860·33-s − 160·35-s − 166·37-s + 1.24e3·39-s + 34·41-s − 946·43-s + 1.12e3·45-s − 234·47-s + 51·49-s − 140·51-s + 1.14e3·53-s + 1.72e3·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s + 1.78·5-s − 0.431·7-s + 2.07·9-s + 2.35·11-s − 2.64·13-s − 3.44·15-s + 0.199·17-s + 1.06·19-s + 0.831·21-s − 0.616·23-s + 0.543·25-s − 1.76·27-s − 0.557·29-s − 1.88·31-s − 4.53·33-s − 0.772·35-s − 0.737·37-s + 5.09·39-s + 0.129·41-s − 3.35·43-s + 3.71·45-s − 0.726·47-s + 0.148·49-s − 0.384·51-s + 2.96·53-s + 4.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 10 T + 44 T^{2} + 128 T^{3} + 44 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 p T + 332 T^{2} - 4266 T^{3} + 332 p^{3} T^{4} - 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 8 T + 13 T^{2} - 10064 T^{3} + 13 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 86 T + 5196 T^{2} - 206216 T^{3} + 5196 p^{3} T^{4} - 86 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 124 T + 868 p T^{2} + 598190 T^{3} + 868 p^{4} T^{4} + 124 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 14 T + 7319 T^{2} + 102732 T^{3} + 7319 p^{3} T^{4} - 14 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 88 T + 21429 T^{2} - 1176992 T^{3} + 21429 p^{3} T^{4} - 88 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 68 T + 3481 T^{2} + 484376 T^{3} + 3481 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 326 T + 122208 T^{2} + 20405404 T^{3} + 122208 p^{3} T^{4} + 326 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 166 T + 97687 T^{2} + 264140 p T^{3} + 97687 p^{3} T^{4} + 166 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 34 T + 105619 T^{2} + 557036 T^{3} + 105619 p^{3} T^{4} - 34 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 22 p T + 528340 T^{2} + 178820552 T^{3} + 528340 p^{3} T^{4} + 22 p^{7} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 234 T + 191424 T^{2} + 40015348 T^{3} + 191424 p^{3} T^{4} + 234 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 1144 T + 785476 T^{2} - 349244770 T^{3} + 785476 p^{3} T^{4} - 1144 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 488 T + 309309 T^{2} - 129083456 T^{3} + 309309 p^{3} T^{4} - 488 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 450 T + 227619 T^{2} + 136349404 T^{3} + 227619 p^{3} T^{4} + 450 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 52 T + 856113 T^{2} + 29295224 T^{3} + 856113 p^{3} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1196 T + 1207697 T^{2} + 825944328 T^{3} + 1207697 p^{3} T^{4} + 1196 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2434 T + 3136427 T^{2} - 2423395588 T^{3} + 3136427 p^{3} T^{4} - 2434 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 742 T + 74784 T^{2} - 287831572 T^{3} + 74784 p^{3} T^{4} + 742 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 464 T + 1010485 T^{2} - 314181968 T^{3} + 1010485 p^{3} T^{4} - 464 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1986 T + 3400803 T^{2} - 3069895316 T^{3} + 3400803 p^{3} T^{4} - 1986 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 406 T + 1652683 T^{2} + 293831804 T^{3} + 1652683 p^{3} T^{4} + 406 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.156774943680153008678750605182, −7.75689715851637566026366610763, −7.69469906199012538756661666341, −7.19770558151881034044899135228, −6.86768113384365264226636555877, −6.82188876342972768188019552604, −6.76415050442791373715543864020, −6.35969162349904090961182456758, −5.94338424998613637787741270272, −5.89163029454459527808970552514, −5.38438234365605865220607644909, −5.27959865578861643792132960197, −5.26190660410232623467982398960, −4.90965782110878653881011399474, −4.48849755835635007050633100605, −3.94678668271016741209032066036, −3.93227634635400021816749345084, −3.47308319686156569495406977651, −3.24036062006839054870593742032, −2.42069375554332488388597857296, −2.33972027897559460446525068928, −1.94170474932744337250049912194, −1.61747383993137737663772829777, −1.13647946646198047682354371983, −1.11336994971282259043804526175, 0, 0, 0,
1.11336994971282259043804526175, 1.13647946646198047682354371983, 1.61747383993137737663772829777, 1.94170474932744337250049912194, 2.33972027897559460446525068928, 2.42069375554332488388597857296, 3.24036062006839054870593742032, 3.47308319686156569495406977651, 3.93227634635400021816749345084, 3.94678668271016741209032066036, 4.48849755835635007050633100605, 4.90965782110878653881011399474, 5.26190660410232623467982398960, 5.27959865578861643792132960197, 5.38438234365605865220607644909, 5.89163029454459527808970552514, 5.94338424998613637787741270272, 6.35969162349904090961182456758, 6.76415050442791373715543864020, 6.82188876342972768188019552604, 6.86768113384365264226636555877, 7.19770558151881034044899135228, 7.69469906199012538756661666341, 7.75689715851637566026366610763, 8.156774943680153008678750605182
