| L(s) = 1 | − 2·4-s + 2·5-s + 9·7-s − 8-s + 2·11-s + 4·13-s − 3·17-s + 7·19-s − 4·20-s − 23-s − 6·25-s − 18·28-s + 11·29-s + 13·31-s + 4·32-s + 18·35-s − 5·37-s − 2·40-s + 41-s + 6·43-s − 4·44-s − 11·47-s + 37·49-s − 8·52-s − 2·53-s + 4·55-s − 9·56-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s + 3.40·7-s − 0.353·8-s + 0.603·11-s + 1.10·13-s − 0.727·17-s + 1.60·19-s − 0.894·20-s − 0.208·23-s − 6/5·25-s − 3.40·28-s + 2.04·29-s + 2.33·31-s + 0.707·32-s + 3.04·35-s − 0.821·37-s − 0.316·40-s + 0.156·41-s + 0.914·43-s − 0.603·44-s − 1.60·47-s + 37/7·49-s − 1.10·52-s − 0.274·53-s + 0.539·55-s − 1.20·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.836616661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.836616661\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 59 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T^{2} + T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 9 T + 44 T^{2} - 142 T^{3} + 44 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 48 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 32 T^{2} - 6 p T^{3} + 32 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 8 T^{2} + 4 T^{3} + 8 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 68 T^{2} - 270 T^{3} + 68 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 42 T^{2} - 18 T^{3} + 42 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 11 T + 96 T^{2} - 564 T^{3} + 96 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 13 T + 130 T^{2} - 778 T^{3} + 130 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 5 T + 92 T^{2} + 384 T^{3} + 92 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 84 T^{2} - 156 T^{3} + 84 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 38 T^{2} + 76 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 11 T + 104 T^{2} + 538 T^{3} + 104 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 2 T + 70 T^{2} + 270 T^{3} + 70 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + T + 82 T^{2} + 220 T^{3} + 82 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 82 T^{2} - 556 T^{3} + 82 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 26 T + 406 T^{2} + 4116 T^{3} + 406 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 7 T + 78 T^{2} - 304 T^{3} + 78 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 206 T^{2} - 348 T^{3} + 206 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 3 T + 50 T^{2} - 350 T^{3} + 50 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 23 T + 358 T^{2} - 3816 T^{3} + 358 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 266 T^{2} - 2514 T^{3} + 266 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.812301984096908176127672789655, −9.159480497709874481609430298033, −8.952125711390761157900536425738, −8.928695111012322781156302266931, −8.379624904921779683492917801739, −8.234828292545763839450003928229, −7.941630945067703886910291904675, −7.77571667724881359329528129032, −7.48809770104953254392174709930, −6.73762326160572739707433437090, −6.51180290175362163669858722021, −6.22334770174103380489192177401, −5.94736659642020166049637041038, −5.20678390383459871958552350532, −5.12249661926270967887039006728, −5.10416370913007519013257111007, −4.40904498867961525995302014919, −4.23734790304655394953437010350, −4.16840493084790723037648916596, −3.21423241276739828873088532527, −2.95921522121758916556389057452, −2.14803695002667900798065678134, −1.88245306324936485300213913762, −1.21382042280888650694757769281, −1.08101838136937823762257877625,
1.08101838136937823762257877625, 1.21382042280888650694757769281, 1.88245306324936485300213913762, 2.14803695002667900798065678134, 2.95921522121758916556389057452, 3.21423241276739828873088532527, 4.16840493084790723037648916596, 4.23734790304655394953437010350, 4.40904498867961525995302014919, 5.10416370913007519013257111007, 5.12249661926270967887039006728, 5.20678390383459871958552350532, 5.94736659642020166049637041038, 6.22334770174103380489192177401, 6.51180290175362163669858722021, 6.73762326160572739707433437090, 7.48809770104953254392174709930, 7.77571667724881359329528129032, 7.941630945067703886910291904675, 8.234828292545763839450003928229, 8.379624904921779683492917801739, 8.928695111012322781156302266931, 8.952125711390761157900536425738, 9.159480497709874481609430298033, 9.812301984096908176127672789655
