| L(s) = 1 | − 20·3-s − 250·5-s − 1.66e3·7-s + 442·9-s + 3.60e3·11-s − 1.31e4·13-s + 5.00e3·15-s + 5.46e3·17-s − 4.04e4·19-s + 3.32e4·21-s + 4.18e4·23-s + 4.68e4·25-s − 5.34e4·27-s − 1.18e5·29-s + 1.15e5·31-s − 7.20e4·33-s + 4.15e5·35-s − 3.06e5·37-s + 2.63e5·39-s − 3.53e5·41-s + 1.21e6·43-s − 1.10e5·45-s + 2.06e6·47-s + 7.85e5·49-s − 1.09e5·51-s − 1.40e6·53-s − 9.00e5·55-s + ⋯ |
| L(s) = 1 | − 0.427·3-s − 0.894·5-s − 1.82·7-s + 0.202·9-s + 0.815·11-s − 1.66·13-s + 0.382·15-s + 0.269·17-s − 1.35·19-s + 0.782·21-s + 0.716·23-s + 3/5·25-s − 0.522·27-s − 0.903·29-s + 0.698·31-s − 0.348·33-s + 1.63·35-s − 0.996·37-s + 0.711·39-s − 0.800·41-s + 2.33·43-s − 0.180·45-s + 2.90·47-s + 0.953·49-s − 0.115·51-s − 1.29·53-s − 0.729·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 20 T - 14 p T^{2} + 20 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 1660 T + 1970190 T^{2} + 1660 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3600 T + 5634742 T^{2} - 3600 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 13180 T + 163072398 T^{2} + 13180 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5460 T - 160982138 T^{2} - 5460 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 40472 T + 2050920774 T^{2} + 40472 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 41820 T + 7228954990 T^{2} - 41820 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4092 p T + 37435002574 T^{2} + 4092 p^{8} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 115928 T + 13575043518 T^{2} - 115928 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 306940 T + 120498758430 T^{2} + 306940 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 353148 T + 314020811638 T^{2} + 353148 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1215340 T + 866800610214 T^{2} - 1215340 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2068500 T + 2082813588382 T^{2} - 2068500 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1400460 T + 2025459025630 T^{2} + 1400460 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1992504 T + 5477605919542 T^{2} + 1992504 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1678676 T + 900787415886 T^{2} - 1678676 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3663940 T + 12525094986870 T^{2} - 3663940 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1794936 T + 17428289847406 T^{2} + 1794936 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5062180 T + 19516555685238 T^{2} - 5062180 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10178224 T + 57668559431262 T^{2} + 10178224 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7214100 T + 67277011758070 T^{2} + 7214100 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15330828 T + 138366637288054 T^{2} + 15330828 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14024020 T + 210744479822310 T^{2} - 14024020 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.14497005441773210277097136658, −9.633175910031166441697865035961, −9.414019794881688312553636429415, −8.892390138050194911728324620326, −8.351291955219212623396378792476, −7.63182288789485044069694415231, −7.07585768730969191188135180480, −7.00052220340748173150260442231, −6.37674130233847782543928296494, −5.83927626245592572145027390645, −5.31730845890010829824765342927, −4.55261253903191741905846657910, −3.99152436356428663101969615901, −3.77811153507082469672327400027, −2.72873742086532963586098316901, −2.70316916940780002485367871472, −1.60289393579971851607442714322, −0.75455877618792294764729260962, 0, 0,
0.75455877618792294764729260962, 1.60289393579971851607442714322, 2.70316916940780002485367871472, 2.72873742086532963586098316901, 3.77811153507082469672327400027, 3.99152436356428663101969615901, 4.55261253903191741905846657910, 5.31730845890010829824765342927, 5.83927626245592572145027390645, 6.37674130233847782543928296494, 7.00052220340748173150260442231, 7.07585768730969191188135180480, 7.63182288789485044069694415231, 8.351291955219212623396378792476, 8.892390138050194911728324620326, 9.414019794881688312553636429415, 9.633175910031166441697865035961, 10.14497005441773210277097136658
