| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s + 2·5-s + 9·6-s − 4·7-s + 10·8-s + 4·9-s + 6·10-s + 3·11-s + 18·12-s − 12·14-s + 6·15-s + 15·16-s + 5·17-s + 12·18-s + 19-s + 12·20-s − 12·21-s + 9·22-s + 30·24-s − 3·25-s + 2·27-s − 24·28-s − 10·29-s + 18·30-s − 16·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s + 0.894·5-s + 3.67·6-s − 1.51·7-s + 3.53·8-s + 4/3·9-s + 1.89·10-s + 0.904·11-s + 5.19·12-s − 3.20·14-s + 1.54·15-s + 15/4·16-s + 1.21·17-s + 2.82·18-s + 0.229·19-s + 2.68·20-s − 2.61·21-s + 1.91·22-s + 6.12·24-s − 3/5·25-s + 0.384·27-s − 4.53·28-s − 1.85·29-s + 3.28·30-s − 2.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38614472 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38614472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.03417951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.03417951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 - p T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 48 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 29 T^{2} - 53 T^{3} + 29 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 5 T + 29 T^{2} - 73 T^{3} + 29 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - T + 41 T^{2} - 51 T^{3} + 41 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 41 T^{2} + 56 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 10 T + 83 T^{2} + 476 T^{3} + 83 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 16 T + 169 T^{2} + 1096 T^{3} + 169 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 14 T + 167 T^{2} + 1092 T^{3} + 167 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 7 T + 109 T^{2} - 483 T^{3} + 109 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 11 T + 125 T^{2} - 945 T^{3} + 125 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 2 T + 105 T^{2} + 180 T^{3} + 105 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 131 T^{2} - 56 T^{3} + 131 p T^{4} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 3 T + 117 T^{2} + 481 T^{3} + 117 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 4 T + 39 T^{2} + 424 T^{3} + 39 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 21 T + 285 T^{2} - 2527 T^{3} + 285 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 12 T + 233 T^{2} + 1712 T^{3} + 233 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - T + 133 T^{2} - 397 T^{3} + 133 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 261 T^{2} + 2612 T^{3} + 261 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + T + 233 T^{2} + 179 T^{3} + 233 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 25 T + 361 T^{2} - 3693 T^{3} + 361 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 23 T + 353 T^{2} - 3579 T^{3} + 353 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−10.30865581930478998889887151898, −9.915236562097292882569326542811, −9.666737236034969927007798825788, −9.459385313557622233111663414695, −9.062958491468611985826427522847, −8.869208608778215051040075433886, −8.565393073771598980332744861645, −7.71721875110513221638948468887, −7.66338797789326504887457943179, −7.24316451134787085022104427598, −7.21249570246910018121195909221, −6.59719341491453911097835242223, −6.20773640184731926247036475596, −5.82916242556054109612144594592, −5.77003133570969417033307855381, −5.32336960815916367068312409920, −4.92477470031522754717671864332, −4.13714419330723691839155457923, −3.97093349011223399172636634021, −3.35450276682853117998654869950, −3.32421227875946250421613054064, −3.25001088281601333018317386302, −2.16895258603392404958940291563, −2.13123084321926059731476644181, −1.56691027510761796953266797785,
1.56691027510761796953266797785, 2.13123084321926059731476644181, 2.16895258603392404958940291563, 3.25001088281601333018317386302, 3.32421227875946250421613054064, 3.35450276682853117998654869950, 3.97093349011223399172636634021, 4.13714419330723691839155457923, 4.92477470031522754717671864332, 5.32336960815916367068312409920, 5.77003133570969417033307855381, 5.82916242556054109612144594592, 6.20773640184731926247036475596, 6.59719341491453911097835242223, 7.21249570246910018121195909221, 7.24316451134787085022104427598, 7.66338797789326504887457943179, 7.71721875110513221638948468887, 8.565393073771598980332744861645, 8.869208608778215051040075433886, 9.062958491468611985826427522847, 9.459385313557622233111663414695, 9.666737236034969927007798825788, 9.915236562097292882569326542811, 10.30865581930478998889887151898
