| L(s) = 1 | + 5·3-s + 2·5-s + 35·7-s − 4·9-s + 28·11-s − 109·13-s + 10·15-s − 123·17-s − 57·19-s + 175·21-s + 193·23-s − 92·25-s + 133·27-s − 297·29-s + 140·31-s + 140·33-s + 70·35-s + 38·37-s − 545·39-s + 736·41-s + 514·43-s − 8·45-s − 134·47-s + 77·49-s − 615·51-s + 311·53-s + 56·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s + 0.178·5-s + 1.88·7-s − 0.148·9-s + 0.767·11-s − 2.32·13-s + 0.172·15-s − 1.75·17-s − 0.688·19-s + 1.81·21-s + 1.74·23-s − 0.735·25-s + 0.947·27-s − 1.90·29-s + 0.811·31-s + 0.738·33-s + 0.338·35-s + 0.168·37-s − 2.23·39-s + 2.80·41-s + 1.82·43-s − 0.0265·45-s − 0.415·47-s + 0.224·49-s − 1.68·51-s + 0.806·53-s + 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.465182700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.465182700\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 5 T + 29 T^{2} - 298 T^{3} + 29 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 96 T^{2} + 1016 T^{3} + 96 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5 p T + 164 p T^{2} - 23983 T^{3} + 164 p^{4} T^{4} - 5 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 28 T + 4182 T^{2} - 74894 T^{3} + 4182 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 109 T + 6999 T^{2} + 365490 T^{3} + 6999 p^{3} T^{4} + 109 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 123 T + 17610 T^{2} + 1215235 T^{3} + 17610 p^{3} T^{4} + 123 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 193 T + 39933 T^{2} - 4449614 T^{3} + 39933 p^{3} T^{4} - 193 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 297 T + 71131 T^{2} + 13319430 T^{3} + 71131 p^{3} T^{4} + 297 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 140 T + 62189 T^{2} - 4673384 T^{3} + 62189 p^{3} T^{4} - 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 38 T + 100099 T^{2} - 7279012 T^{3} + 100099 p^{3} T^{4} - 38 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 736 T + 354551 T^{2} - 108718336 T^{3} + 354551 p^{3} T^{4} - 736 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 514 T + 224130 T^{2} - 56635248 T^{3} + 224130 p^{3} T^{4} - 514 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 134 T + 8542 T^{2} - 31064212 T^{3} + 8542 p^{3} T^{4} + 134 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 311 T + 285651 T^{2} - 106221286 T^{3} + 285651 p^{3} T^{4} - 311 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 199 T + 35145 T^{2} - 35868602 T^{3} + 35145 p^{3} T^{4} + 199 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 56 T + 192812 T^{2} + 95078710 T^{3} + 192812 p^{3} T^{4} - 56 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 509 T + 553245 T^{2} - 128354670 T^{3} + 553245 p^{3} T^{4} - 509 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 874 T + 862737 T^{2} - 513398212 T^{3} + 862737 p^{3} T^{4} - 874 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 203 T + 1104798 T^{2} - 157466799 T^{3} + 1104798 p^{3} T^{4} - 203 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 242 T + 502613 T^{2} + 37031420 T^{3} + 502613 p^{3} T^{4} + 242 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 62 T + 1258913 T^{2} - 154550580 T^{3} + 1258913 p^{3} T^{4} - 62 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1764 T + 3042123 T^{2} - 2614453128 T^{3} + 3042123 p^{3} T^{4} - 1764 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2178 T + 3988819 T^{2} + 4075506692 T^{3} + 3988819 p^{3} T^{4} + 2178 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.846223777462568546314360223324, −9.480228984253261288386880547318, −9.388831197020595096584902351697, −9.176985651202488577271883588222, −8.654542548338446332885722090904, −8.541089446182544506429539718246, −8.182660064137815763193007904729, −7.54733890620872877843504723655, −7.53842227156571053093163280714, −7.45115180981913637410561665473, −6.68603890120793934962496917906, −6.44266419423815731733596920049, −6.18254354084535631253989721489, −5.31019784896800884092238391877, −5.06480889553315385568676961501, −5.04918589113692540660984785140, −4.30108937850090138001115544203, −4.21820289367402961464997567022, −3.79377537587383463836047517010, −2.82215607317539289441585266022, −2.49607769856702221341471882810, −2.41572612669935721630567134407, −1.83603365375001199738450358149, −1.17861291637460230647129740104, −0.45539880345220226819484374287,
0.45539880345220226819484374287, 1.17861291637460230647129740104, 1.83603365375001199738450358149, 2.41572612669935721630567134407, 2.49607769856702221341471882810, 2.82215607317539289441585266022, 3.79377537587383463836047517010, 4.21820289367402961464997567022, 4.30108937850090138001115544203, 5.04918589113692540660984785140, 5.06480889553315385568676961501, 5.31019784896800884092238391877, 6.18254354084535631253989721489, 6.44266419423815731733596920049, 6.68603890120793934962496917906, 7.45115180981913637410561665473, 7.53842227156571053093163280714, 7.54733890620872877843504723655, 8.182660064137815763193007904729, 8.541089446182544506429539718246, 8.654542548338446332885722090904, 9.176985651202488577271883588222, 9.388831197020595096584902351697, 9.480228984253261288386880547318, 9.846223777462568546314360223324
