| L(s) = 1 | + 2·2-s − 2·3-s − 4-s + 4·5-s − 4·6-s + 3·7-s − 5·8-s − 4·9-s + 8·10-s + 8·11-s + 2·12-s + 6·14-s − 8·15-s − 16-s − 2·17-s − 8·18-s + 4·19-s − 4·20-s − 6·21-s + 16·22-s − 5·23-s + 10·24-s − 2·25-s + 13·27-s − 3·28-s − 29-s − 16·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s − 1/2·4-s + 1.78·5-s − 1.63·6-s + 1.13·7-s − 1.76·8-s − 4/3·9-s + 2.52·10-s + 2.41·11-s + 0.577·12-s + 1.60·14-s − 2.06·15-s − 1/4·16-s − 0.485·17-s − 1.88·18-s + 0.917·19-s − 0.894·20-s − 1.30·21-s + 3.41·22-s − 1.04·23-s + 2.04·24-s − 2/5·25-s + 2.50·27-s − 0.566·28-s − 0.185·29-s − 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4826809 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4826809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.937033620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.937033620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - p T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 + 2 T + 8 T^{2} + 11 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 4 T + 18 T^{2} - 39 T^{3} + 18 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 17 T^{2} - 29 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 8 T + 52 T^{2} - 189 T^{3} + 52 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 36 T^{2} + 81 T^{3} + 36 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 4 T + 46 T^{2} - 151 T^{3} + 46 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 5 T + 68 T^{2} + 217 T^{3} + 68 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + T + 43 T^{2} + 141 T^{3} + 43 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 57 T^{2} - 143 T^{3} + 57 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 12 T + 152 T^{2} + 917 T^{3} + 152 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 7 T + 74 T^{2} - 623 T^{3} + 74 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 13 T + 169 T^{2} - 1105 T^{3} + 169 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 18 T + 242 T^{2} - 1859 T^{3} + 242 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - T + 73 T^{2} + 231 T^{3} + 73 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 19 T + 260 T^{2} - 2243 T^{3} + 260 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 4 T + 116 T^{2} - 249 T^{3} + 116 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - T + 129 T^{2} - 175 T^{3} + 129 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 27 T + 435 T^{2} - 4381 T^{3} + 435 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 9 T + 99 T^{2} + 403 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 5 T + 75 T^{2} + 917 T^{3} + 75 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 7 T + 109 T^{2} - 1365 T^{3} + 109 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 11 T + 193 T^{2} - 1677 T^{3} + 193 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 7 T + 207 T^{2} + 1057 T^{3} + 207 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−11.68893853811034471120932270755, −11.33150064057715265287341268911, −10.85100625135482085432967389596, −10.74836176683893083695490809970, −9.921099931600111742647015868312, −9.850812012320423426642006832263, −9.505382837073334992884561685399, −8.879030416311750200249599743779, −8.816353062533442556770931127476, −8.757080549691610901615934076372, −7.83292919661397177958864491463, −7.69632967238829786365417352756, −6.87114205305089699269934736758, −6.34523874613162789796352729370, −6.11296835528273051749193674268, −5.97582401870146063468266345425, −5.27059605891272504712813429219, −5.26167575173753922282991782557, −5.15282644837173323114009188850, −4.17900829812228157505087814893, −3.86542499175143892522610473183, −3.82637468892358221710217802018, −2.58504684420432331672462752832, −2.07638593057499280393476465311, −1.10970842323437459325835868329,
1.10970842323437459325835868329, 2.07638593057499280393476465311, 2.58504684420432331672462752832, 3.82637468892358221710217802018, 3.86542499175143892522610473183, 4.17900829812228157505087814893, 5.15282644837173323114009188850, 5.26167575173753922282991782557, 5.27059605891272504712813429219, 5.97582401870146063468266345425, 6.11296835528273051749193674268, 6.34523874613162789796352729370, 6.87114205305089699269934736758, 7.69632967238829786365417352756, 7.83292919661397177958864491463, 8.757080549691610901615934076372, 8.816353062533442556770931127476, 8.879030416311750200249599743779, 9.505382837073334992884561685399, 9.850812012320423426642006832263, 9.921099931600111742647015868312, 10.74836176683893083695490809970, 10.85100625135482085432967389596, 11.33150064057715265287341268911, 11.68893853811034471120932270755
