| L(s) = 1 | + 3·3-s − 5-s + 9·7-s + 6·9-s + 5·11-s − 3·15-s − 8·17-s − 4·19-s + 27·21-s + 2·25-s + 10·27-s + 11·29-s + 5·31-s + 15·33-s − 9·35-s + 8·37-s − 2·41-s − 12·43-s − 6·45-s + 4·47-s + 40·49-s − 24·51-s + 5·53-s − 5·55-s − 12·57-s − 5·59-s + 22·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 3.40·7-s + 2·9-s + 1.50·11-s − 0.774·15-s − 1.94·17-s − 0.917·19-s + 5.89·21-s + 2/5·25-s + 1.92·27-s + 2.04·29-s + 0.898·31-s + 2.61·33-s − 1.52·35-s + 1.31·37-s − 0.312·41-s − 1.82·43-s − 0.894·45-s + 0.583·47-s + 40/7·49-s − 3.36·51-s + 0.686·53-s − 0.674·55-s − 1.58·57-s − 0.650·59-s + 2.81·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(25.99205603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(25.99205603\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + T - T^{2} - 19 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 9 T + 41 T^{2} - 125 T^{3} + 41 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 25 T^{2} - 111 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 8 T + 63 T^{2} + 264 T^{3} + 63 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 88 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 41 T^{2} - 56 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 11 T + 111 T^{2} - 21 p T^{3} + 111 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 43 T^{2} - 185 T^{3} + 43 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 8 T + 67 T^{2} - 584 T^{3} + 67 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 2 T + 59 T^{2} - 68 T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 12 T + 149 T^{2} + 928 T^{3} + 149 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 4 T + 109 T^{2} - 312 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 5 T + 123 T^{2} - 573 T^{3} + 123 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 5 T + 141 T^{2} + 423 T^{3} + 141 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 22 T + 335 T^{2} - 3012 T^{3} + 335 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 185 T^{2} - 700 T^{3} + 185 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 18 T + 293 T^{2} + 2548 T^{3} + 293 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 13 T + 189 T^{2} - 1885 T^{3} + 189 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 31 T + 513 T^{2} + 5431 T^{3} + 513 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 13 T + 121 T^{2} - 591 T^{3} + 121 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 14 T + 211 T^{2} - 2436 T^{3} + 211 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 23 T + 381 T^{2} - 47 p T^{3} + 381 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.07407645853732151939640425788, −6.79422148340942119837430215179, −6.54983483781620696437646686230, −6.38332040132208548899647489374, −6.02203917093923973386319320259, −5.71288012745902610938381863789, −5.49899560743922865113482615454, −4.92686729235372645666758831862, −4.80238638213805621284302924001, −4.62949765483961591500623942103, −4.56685267715797850881826346700, −4.32634901193769798644573483481, −4.22600850872310321550698232775, −3.71410816135174402760476839648, −3.60583307969797927715169294998, −3.31277400546431827346903232661, −2.70325707116672335674925087520, −2.69471139406962644985427789637, −2.35546743651468869400597329739, −1.89397030075278217426767582139, −1.81769533703880865270978632236, −1.76663806319342905446693793824, −1.19200341787482235924645974352, −0.77821818188097693257205406722, −0.71025987110883785642170192373,
0.71025987110883785642170192373, 0.77821818188097693257205406722, 1.19200341787482235924645974352, 1.76663806319342905446693793824, 1.81769533703880865270978632236, 1.89397030075278217426767582139, 2.35546743651468869400597329739, 2.69471139406962644985427789637, 2.70325707116672335674925087520, 3.31277400546431827346903232661, 3.60583307969797927715169294998, 3.71410816135174402760476839648, 4.22600850872310321550698232775, 4.32634901193769798644573483481, 4.56685267715797850881826346700, 4.62949765483961591500623942103, 4.80238638213805621284302924001, 4.92686729235372645666758831862, 5.49899560743922865113482615454, 5.71288012745902610938381863789, 6.02203917093923973386319320259, 6.38332040132208548899647489374, 6.54983483781620696437646686230, 6.79422148340942119837430215179, 7.07407645853732151939640425788
