| L(s) = 1 | − 20·3-s + 74·5-s + 243·9-s − 124·11-s + 956·13-s − 1.48e3·15-s + 1.19e3·17-s − 3.04e3·19-s − 184·23-s + 3.12e3·25-s − 6.58e3·27-s − 6.56e3·29-s + 5.72e3·31-s + 2.48e3·33-s − 1.03e4·37-s − 1.91e4·39-s − 1.77e4·41-s − 1.83e4·43-s + 1.79e4·45-s − 2.36e4·47-s − 2.39e4·51-s − 1.16e4·53-s − 9.17e3·55-s + 6.08e4·57-s − 1.68e4·59-s + 1.84e4·61-s + 7.07e4·65-s + ⋯ |
| L(s) = 1 | − 1.28·3-s + 1.32·5-s + 9-s − 0.308·11-s + 1.56·13-s − 1.69·15-s + 1.00·17-s − 1.93·19-s − 0.0725·23-s + 25-s − 1.73·27-s − 1.44·29-s + 1.07·31-s + 0.396·33-s − 1.24·37-s − 2.01·39-s − 1.65·41-s − 1.51·43-s + 1.32·45-s − 1.56·47-s − 1.28·51-s − 0.571·53-s − 0.409·55-s + 2.48·57-s − 0.631·59-s + 0.635·61-s + 2.07·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4049739436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4049739436\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 20 T + 157 T^{2} + 20 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 74 T + 2351 T^{2} - 74 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 124 T - 145675 T^{2} + 124 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 478 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1198 T + 15347 T^{2} - 1198 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3044 T + 6789837 T^{2} + 3044 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 p T - 12103 p^{2} T^{2} + 8 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3282 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5728 T + 4180833 T^{2} - 5728 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 10326 T + 37282319 T^{2} + 10326 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8886 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9188 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 23664 T + 330639889 T^{2} + 23664 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 11686 T - 281632897 T^{2} + 11686 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 16876 T - 430124923 T^{2} + 16876 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 18482 T - 503011977 T^{2} - 18482 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15532 T - 1108882083 T^{2} - 15532 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 31960 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4886 T - 2049198597 T^{2} - 4886 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 44560 T - 1091462799 T^{2} + 44560 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 67364 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 71994 T - 400923413 T^{2} + 71994 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 48866 T + p^{5} T^{2} )^{2} \) |
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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
−10.66783080981340879450835246345, −10.40915354379092878042248877332, −9.746723394328253757376668773537, −9.742208080224699266486639568271, −8.814920390660059116567004800771, −8.494448480279931156470561034938, −8.031998525188269523348244656781, −7.30518112164259140487893381265, −6.54713160563795835363331232449, −6.42987583563835415425886580191, −5.97738547920479228135702661687, −5.49035294308262908639791900610, −5.10952667923917946020897897230, −4.52794224933280397978147364901, −3.65707688917403336997711861529, −3.33674395205035168783289064063, −2.17455745708859994782932089886, −1.65532519806933513359798353841, −1.30409758816197589469217384022, −0.16995465515635073577894450080,
0.16995465515635073577894450080, 1.30409758816197589469217384022, 1.65532519806933513359798353841, 2.17455745708859994782932089886, 3.33674395205035168783289064063, 3.65707688917403336997711861529, 4.52794224933280397978147364901, 5.10952667923917946020897897230, 5.49035294308262908639791900610, 5.97738547920479228135702661687, 6.42987583563835415425886580191, 6.54713160563795835363331232449, 7.30518112164259140487893381265, 8.031998525188269523348244656781, 8.494448480279931156470561034938, 8.814920390660059116567004800771, 9.742208080224699266486639568271, 9.746723394328253757376668773537, 10.40915354379092878042248877332, 10.66783080981340879450835246345
