| L(s) = 1 | + 8·2-s + 18·3-s + 48·4-s + 18·5-s + 144·6-s + 256·8-s + 243·9-s + 144·10-s + 2·11-s + 864·12-s + 288·13-s + 324·15-s + 1.28e3·16-s + 1.53e3·17-s + 1.94e3·18-s + 1.18e3·19-s + 864·20-s + 16·22-s + 3.39e3·23-s + 4.60e3·24-s − 1.30e3·25-s + 2.30e3·26-s + 2.91e3·27-s − 3.97e3·29-s + 2.59e3·30-s + 7.59e3·31-s + 6.14e3·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.321·5-s + 1.63·6-s + 1.41·8-s + 9-s + 0.455·10-s + 0.00498·11-s + 1.73·12-s + 0.472·13-s + 0.371·15-s + 5/4·16-s + 1.28·17-s + 1.41·18-s + 0.754·19-s + 0.482·20-s + 0.00704·22-s + 1.33·23-s + 1.63·24-s − 0.416·25-s + 0.668·26-s + 0.769·27-s − 0.877·29-s + 0.525·30-s + 1.41·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(19.62804613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.62804613\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 18 T + 1626 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 59002 T^{2} - 2 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 288 T + 688042 T^{2} - 288 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 90 p T + 2366314 T^{2} - 90 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1188 T + 4834534 T^{2} - 1188 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3390 T + 6218086 T^{2} - 3390 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3976 T + 31254662 T^{2} + 3976 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7596 T + 61727326 T^{2} - 7596 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2688 T + 126774470 T^{2} - 2688 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 36630 T + 559995322 T^{2} - 36630 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 23032 T + 329072262 T^{2} + 23032 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 864 T + 46643358 T^{2} - 864 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 32920 T + 983844566 T^{2} + 32920 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 26712 T + 697343334 T^{2} + 26712 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20412 T + 1230091258 T^{2} - 20412 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 36172 T + 33148290 p T^{2} + 36172 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 73706 T + 3356433686 T^{2} - 73706 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 74772 T + 3993671602 T^{2} + 74772 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 23116 T + 6249589662 T^{2} + 23116 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 147816 T + 13316083030 T^{2} - 147816 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 164646 T + 17878567722 T^{2} - 164646 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 162036 T + 20528488258 T^{2} - 162036 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.03126449600863208109055982126, −11.00471394897125006080699569534, −10.21015340067008375016439628505, −9.724665766568722305175836917371, −9.356517174017467275586622541514, −8.849245838852459243027358653511, −8.030569639097068673788683219051, −7.73122648234537336719519786689, −7.38916085927152052026579388151, −6.65197399475395838663448612034, −6.08625616193183677395384595949, −5.73821327159221643919228818660, −4.79343678164705587341720345002, −4.74491002185730943442141796740, −3.63689787001847966387176938758, −3.49898359280308871950861336258, −2.80945490551206674681251538260, −2.32358838506610979906653471636, −1.40505268813211093398721159124, −0.950314725667538151007569784201,
0.950314725667538151007569784201, 1.40505268813211093398721159124, 2.32358838506610979906653471636, 2.80945490551206674681251538260, 3.49898359280308871950861336258, 3.63689787001847966387176938758, 4.74491002185730943442141796740, 4.79343678164705587341720345002, 5.73821327159221643919228818660, 6.08625616193183677395384595949, 6.65197399475395838663448612034, 7.38916085927152052026579388151, 7.73122648234537336719519786689, 8.030569639097068673788683219051, 8.849245838852459243027358653511, 9.356517174017467275586622541514, 9.724665766568722305175836917371, 10.21015340067008375016439628505, 11.00471394897125006080699569534, 11.03126449600863208109055982126
