| L(s) = 1 | − 46·13-s − 50·25-s − 146·37-s + 23·49-s + 94·61-s − 286·73-s + 338·97-s + 428·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.24e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.53·13-s − 2·25-s − 3.94·37-s + 0.469·49-s + 1.54·61-s − 3.91·73-s + 3.48·97-s + 3.92·109-s + 2·121-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 + 7.39·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4403844775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4403844775\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 143 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 169 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.51179405273363309167022656763, −10.29242288483666051886795119982, −10.22152870570424121107761383439, −10.04424404343104830137444902810, −9.481233328206213933275399165178, −8.751777695806891265971437499754, −8.693848170766365944579944699662, −7.65095764115578150809566188369, −7.48241099712259750297486141231, −7.17118697576071707430259826945, −6.66300592124507198180826828130, −5.74099678926026820852066880323, −5.48436683542902779423348799724, −4.74006845062825056818415390416, −4.60970152247733536983135654892, −3.63364292158129524732843799587, −3.08541705754602187038259046215, −2.07565707218682719379830347463, −2.03029521795589968812532915905, −0.25987078752985166652579883293,
0.25987078752985166652579883293, 2.03029521795589968812532915905, 2.07565707218682719379830347463, 3.08541705754602187038259046215, 3.63364292158129524732843799587, 4.60970152247733536983135654892, 4.74006845062825056818415390416, 5.48436683542902779423348799724, 5.74099678926026820852066880323, 6.66300592124507198180826828130, 7.17118697576071707430259826945, 7.48241099712259750297486141231, 7.65095764115578150809566188369, 8.693848170766365944579944699662, 8.751777695806891265971437499754, 9.481233328206213933275399165178, 10.04424404343104830137444902810, 10.22152870570424121107761383439, 10.29242288483666051886795119982, 11.51179405273363309167022656763
