| L(s) = 1 | − 1.95e3·8-s + 8.39e5·23-s + 1.17e6·25-s − 1.06e6·27-s + 1.72e7·49-s + 4.58e7·59-s − 1.29e7·64-s − 4.70e8·101-s + 6.43e8·121-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 − 1.63e9·184-s + 191-s + 193-s + 197-s + 199-s − 2.28e9·200-s + ⋯ |
| L(s) = 1 | − 0.476·8-s + 3·23-s + 3·25-s − 1.99·27-s + 3·49-s + 3.78·59-s − 0.773·64-s − 4.52·101-s + 3·121-s − 1.42·184-s − 1.42·200-s + 0.952·216-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s+4)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.101080880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.101080880\) |
| \(L(5)\) |
|
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 | $C_1$ | \( ( 1 - p^{4} T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + 1951 T^{3} + p^{24} T^{6} \) |
| 3 | $D_{6}$ | \( 1 + 1062686 T^{3} + p^{24} T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 13 | $D_{6}$ | \( 1 - 25363320370274 T^{3} + p^{24} T^{6} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 29 | $D_{6}$ | \( 1 + 647932355939762206 T^{3} + p^{24} T^{6} \) |
| 31 | $D_{6}$ | \( 1 - 1237087799571624194 T^{3} + p^{24} T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 41 | $D_{6}$ | \( 1 - 12576527614080568514 T^{3} + p^{24} T^{6} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 47 | $D_{6}$ | \( 1 - \)\(19\!\cdots\!54\)\( T^{3} + p^{24} T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 59 | $C_2$ | \( ( 1 - 15279074 T + p^{8} T^{2} )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 71 | $D_{6}$ | \( 1 - \)\(28\!\cdots\!94\)\( T^{3} + p^{24} T^{6} \) |
| 73 | $D_{6}$ | \( 1 - \)\(39\!\cdots\!54\)\( T^{3} + p^{24} T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3} \) |
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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
−14.60505482600838178979882243183, −13.57754776366493196214640179035, −13.57358699137694629103351250428, −12.94340974939646264484421352028, −12.58994809827745075575670473695, −12.20573812798288630229092135669, −11.48934113174410361454083235145, −11.13020128337881722518285137089, −10.78407804078334704513648951630, −10.26071600882990030330086952673, −9.569195258661682746768575480396, −9.047364399505374511119356771709, −8.785528370880907108738112345945, −8.312193720324238660650046295629, −7.22331694672823028526268534820, −7.12821196931798717727353152062, −6.61290661913085672954290949270, −5.48365710965517680196720795192, −5.41807275592759405744549257376, −4.54563824817196368509283972522, −3.74241792710594257007497295027, −2.95312795013608813421629405734, −2.44985366376325530211392602373, −1.15438384073458975501791693746, −0.66124939014099614407429649788,
0.66124939014099614407429649788, 1.15438384073458975501791693746, 2.44985366376325530211392602373, 2.95312795013608813421629405734, 3.74241792710594257007497295027, 4.54563824817196368509283972522, 5.41807275592759405744549257376, 5.48365710965517680196720795192, 6.61290661913085672954290949270, 7.12821196931798717727353152062, 7.22331694672823028526268534820, 8.312193720324238660650046295629, 8.785528370880907108738112345945, 9.047364399505374511119356771709, 9.569195258661682746768575480396, 10.26071600882990030330086952673, 10.78407804078334704513648951630, 11.13020128337881722518285137089, 11.48934113174410361454083235145, 12.20573812798288630229092135669, 12.58994809827745075575670473695, 12.94340974939646264484421352028, 13.57358699137694629103351250428, 13.57754776366493196214640179035, 14.60505482600838178979882243183
