| L(s) = 1 | + 2-s − 4-s + 5·5-s − 3·7-s + 5·10-s + 2·11-s + 3·13-s − 3·14-s + 16-s + 12·17-s − 3·19-s − 5·20-s + 2·22-s + 8·25-s + 3·26-s + 3·28-s − 29-s − 3·31-s − 2·32-s + 12·34-s − 15·35-s − 3·37-s − 3·38-s + 22·41-s − 3·43-s − 2·44-s + 9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 2.23·5-s − 1.13·7-s + 1.58·10-s + 0.603·11-s + 0.832·13-s − 0.801·14-s + 1/4·16-s + 2.91·17-s − 0.688·19-s − 1.11·20-s + 0.426·22-s + 8/5·25-s + 0.588·26-s + 0.566·28-s − 0.185·29-s − 0.538·31-s − 0.353·32-s + 2.05·34-s − 2.53·35-s − 0.493·37-s − 0.486·38-s + 3.43·41-s − 0.457·43-s − 0.301·44-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.092987739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.092987739\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p T^{2} - 3 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - p T + 17 T^{2} - 39 T^{3} + 17 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 14 T^{2} + 3 T^{3} + 14 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 12 T + 90 T^{2} - 435 T^{3} + 90 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 107 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 36 T^{2} + 9 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 83 T^{2} + 57 T^{3} + 83 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 213 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 57 T^{2} + 303 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 22 T + 278 T^{2} - 2157 T^{3} + 278 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 63 T^{2} + 379 T^{3} + 63 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 87 T^{2} - 657 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 18 T + 234 T^{2} - 1917 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 9 T + 171 T^{2} - 999 T^{3} + 171 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 162 T^{2} + 665 T^{3} + 162 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T^{2} + 683 T^{3} - 6 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 681 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 189 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 288 T^{2} - 2019 T^{3} + 288 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 116 T^{2} - 735 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 3 T + 177 T^{2} + 21 T^{3} + 177 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.667202982911172467936908893282, −9.245579797189346635830173384999, −9.114854047520996330393765466281, −9.100566390312128371793766705955, −8.369818529791513840132380069056, −8.145175021850109391241477118141, −7.76472851329647239949552965129, −7.26844618787575647067209116877, −7.24642224008905757908686137464, −6.64838007076178314778551895602, −6.26928876126177547769500149104, −6.04365020176797562726267992751, −5.86585716091023734608886852475, −5.51514315585852321213174152316, −5.29401098218309369224960252406, −5.11473368907394368245575790124, −4.20049529374879075425604824443, −4.00240543875989882502090043562, −3.92990874286156576178906274739, −3.30014270255967393831870514018, −2.93610755545442122324175624059, −2.29555711549188345684187135350, −2.10742496733941490908995335729, −1.19317538245399821070347558123, −1.01347993391273402801437244725,
1.01347993391273402801437244725, 1.19317538245399821070347558123, 2.10742496733941490908995335729, 2.29555711549188345684187135350, 2.93610755545442122324175624059, 3.30014270255967393831870514018, 3.92990874286156576178906274739, 4.00240543875989882502090043562, 4.20049529374879075425604824443, 5.11473368907394368245575790124, 5.29401098218309369224960252406, 5.51514315585852321213174152316, 5.86585716091023734608886852475, 6.04365020176797562726267992751, 6.26928876126177547769500149104, 6.64838007076178314778551895602, 7.24642224008905757908686137464, 7.26844618787575647067209116877, 7.76472851329647239949552965129, 8.145175021850109391241477118141, 8.369818529791513840132380069056, 9.100566390312128371793766705955, 9.114854047520996330393765466281, 9.245579797189346635830173384999, 9.667202982911172467936908893282
