| L(s) = 1 | − 34·9-s + 1.14e3·17-s + 1.25e3·25-s − 2.49e3·41-s + 4.80e3·49-s − 1.90e4·73-s − 5.40e3·81-s − 1.09e4·89-s + 1.99e4·97-s − 3.18e4·113-s + 2.71e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.90e4·153-s + 157-s + 163-s + 167-s + 5.71e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 0.419·9-s + 3.97·17-s + 2·25-s − 1.48·41-s + 2·49-s − 3.56·73-s − 0.823·81-s − 1.38·89-s + 2.12·97-s − 2.49·113-s + 1.85·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s − 1.66·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 2·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.794779084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.794779084\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 34 T^{2} + p^{8} T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 27166 T^{2} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 574 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 72286 T^{2} + p^{8} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 1246 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 5426402 T^{2} + p^{8} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 24178078 T^{2} + p^{8} T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13944286 T^{2} + p^{8} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9506 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30209954 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5474 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 9982 T + p^{4} T^{2} )^{2} \) |
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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
−12.67717486293940762521249262361, −12.42996047022723421171502658459, −11.78787757824271571237520041419, −11.66478244637359263103149000486, −10.56513950201104980424370549992, −10.35310224306525109383658883399, −9.926252848461247798297904105085, −9.207701164436918721413597116366, −8.578921073190266735569617125057, −8.150598931653641630787758155365, −7.37937395816449745759902880895, −7.14294525900117701021896748558, −6.10359331481474250132689531372, −5.52444890318812891473553435481, −5.20262718412148479899981291715, −4.19172048248693504434207361539, −3.12115948007438289418047758471, −3.05905186407256623706894408065, −1.47032999409989315865597013451, −0.78129426580279171719357823896,
0.78129426580279171719357823896, 1.47032999409989315865597013451, 3.05905186407256623706894408065, 3.12115948007438289418047758471, 4.19172048248693504434207361539, 5.20262718412148479899981291715, 5.52444890318812891473553435481, 6.10359331481474250132689531372, 7.14294525900117701021896748558, 7.37937395816449745759902880895, 8.150598931653641630787758155365, 8.578921073190266735569617125057, 9.207701164436918721413597116366, 9.926252848461247798297904105085, 10.35310224306525109383658883399, 10.56513950201104980424370549992, 11.66478244637359263103149000486, 11.78787757824271571237520041419, 12.42996047022723421171502658459, 12.67717486293940762521249262361
