| L(s) = 1 | − 7·8-s − 69·23-s + 75·25-s − 38·27-s + 147·49-s + 78·59-s − 15·64-s − 498·101-s + 363·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 483·184-s + 191-s + 193-s + 197-s + 199-s − 525·200-s + ⋯ |
| L(s) = 1 | − 7/8·8-s − 3·23-s + 3·25-s − 1.40·27-s + 3·49-s + 1.32·59-s − 0.234·64-s − 4.93·101-s + 3·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 21/8·184-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s − 2.62·200-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s+1)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7038092951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7038092951\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 23 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + 7 T^{3} + p^{6} T^{6} \) |
| 3 | $D_{6}$ | \( 1 + 38 T^{3} + p^{6} T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 13 | $D_{6}$ | \( 1 - 1082 T^{3} + p^{6} T^{6} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 29 | $D_{6}$ | \( 1 - 30746 T^{3} + p^{6} T^{6} \) |
| 31 | $D_{6}$ | \( 1 - 58754 T^{3} + p^{6} T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 41 | $D_{6}$ | \( 1 - 43634 T^{3} + p^{6} T^{6} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 47 | $D_{6}$ | \( 1 + 205342 T^{3} + p^{6} T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 59 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 71 | $D_{6}$ | \( 1 - 667154 T^{3} + p^{6} T^{6} \) |
| 73 | $D_{6}$ | \( 1 - 725042 T^{3} + p^{6} T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
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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
−16.25266090259699302544844085089, −15.38379949405769162888300515781, −15.25045957856847525792594451566, −14.82823558841975961978977417582, −14.30448934002337246511345373331, −13.87709638210915675759205960405, −13.54348074276356119184964203974, −12.88326648893801043752994094227, −12.34454102311100891116481500791, −12.18652506141296656813498136593, −11.66311862169944965457286273302, −11.12595221769137640392299636957, −10.35740039286693641585425892371, −10.33664492004809423462870333956, −9.319600292021350002126565316276, −9.255683549229959107838401341587, −8.254627656874356514129115296000, −8.249320660797426125572295542019, −7.15878206375237697440477144189, −6.81039803586467718459107985552, −5.79533420479259215020430426704, −5.67469663404156687062790353124, −4.46007887250445186823029671800, −3.69917499700457382202237665871, −2.50316094689835549482408964468,
2.50316094689835549482408964468, 3.69917499700457382202237665871, 4.46007887250445186823029671800, 5.67469663404156687062790353124, 5.79533420479259215020430426704, 6.81039803586467718459107985552, 7.15878206375237697440477144189, 8.249320660797426125572295542019, 8.254627656874356514129115296000, 9.255683549229959107838401341587, 9.319600292021350002126565316276, 10.33664492004809423462870333956, 10.35740039286693641585425892371, 11.12595221769137640392299636957, 11.66311862169944965457286273302, 12.18652506141296656813498136593, 12.34454102311100891116481500791, 12.88326648893801043752994094227, 13.54348074276356119184964203974, 13.87709638210915675759205960405, 14.30448934002337246511345373331, 14.82823558841975961978977417582, 15.25045957856847525792594451566, 15.38379949405769162888300515781, 16.25266090259699302544844085089
