| L(s) = 1 | − 5·5-s + 3·7-s + 2·11-s + 3·13-s − 12·17-s + 3·19-s + 8·25-s + 29-s + 3·31-s − 15·35-s − 3·37-s − 22·41-s + 3·43-s + 9·47-s + 6·49-s − 18·53-s − 10·55-s + 9·59-s − 6·61-s − 15·65-s − 9·71-s + 3·73-s + 6·77-s − 15·79-s + 12·83-s + 60·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 2.23·5-s + 1.13·7-s + 0.603·11-s + 0.832·13-s − 2.91·17-s + 0.688·19-s + 8/5·25-s + 0.185·29-s + 0.538·31-s − 2.53·35-s − 0.493·37-s − 3.43·41-s + 0.457·43-s + 1.31·47-s + 6/7·49-s − 2.47·53-s − 1.34·55-s + 1.17·59-s − 0.768·61-s − 1.86·65-s − 1.06·71-s + 0.351·73-s + 0.683·77-s − 1.68·79-s + 1.31·83-s + 6.50·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 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
−7.23516457147933117641356212681, −6.98919547102430271841400297629, −6.70925638682355619871445218229, −6.56756885795437889178425744036, −6.27378151503596418886592598481, −6.24026906498255735895511228893, −5.86305945153960612540258041453, −5.31036803605481234677112997051, −5.20339155961942911440644235514, −5.14583878502827686351705853615, −4.67899439831853684315854346705, −4.53945306419445351133882879308, −4.34892430606718122405381854436, −4.00092919475072741814106512793, −3.89775596988932991323447640733, −3.86523596296885761171498942116, −3.25336880004378211970048356079, −3.08882304553619721310908113002, −3.07037556287849300601877411831, −2.33810364172576856537893144551, −2.19580912396333726176444524810, −1.93340850230431137953131829869, −1.51665031095522094948295371284, −1.09005067207007011361079920650, −1.06710353288087361813254503974, 0, 0, 0,
1.06710353288087361813254503974, 1.09005067207007011361079920650, 1.51665031095522094948295371284, 1.93340850230431137953131829869, 2.19580912396333726176444524810, 2.33810364172576856537893144551, 3.07037556287849300601877411831, 3.08882304553619721310908113002, 3.25336880004378211970048356079, 3.86523596296885761171498942116, 3.89775596988932991323447640733, 4.00092919475072741814106512793, 4.34892430606718122405381854436, 4.53945306419445351133882879308, 4.67899439831853684315854346705, 5.14583878502827686351705853615, 5.20339155961942911440644235514, 5.31036803605481234677112997051, 5.86305945153960612540258041453, 6.24026906498255735895511228893, 6.27378151503596418886592598481, 6.56756885795437889178425744036, 6.70925638682355619871445218229, 6.98919547102430271841400297629, 7.23516457147933117641356212681
