| L(s) = 1 | + 2·4-s + 3·5-s − 7-s + 6·11-s + 4·13-s − 6·17-s + 4·19-s + 6·20-s − 6·23-s + 5·25-s − 2·28-s − 6·29-s + 4·31-s − 3·35-s − 14·37-s − 3·41-s + 43-s + 12·44-s + 9·47-s + 8·52-s + 12·53-s + 18·55-s + 9·59-s + 10·61-s − 8·64-s + 12·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 4-s + 1.34·5-s − 0.377·7-s + 1.80·11-s + 1.10·13-s − 1.45·17-s + 0.917·19-s + 1.34·20-s − 1.25·23-s + 25-s − 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.507·35-s − 2.30·37-s − 0.468·41-s + 0.152·43-s + 1.80·44-s + 1.31·47-s + 1.10·52-s + 1.64·53-s + 2.42·55-s + 1.17·59-s + 1.28·61-s − 64-s + 1.48·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.312955289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.312955289\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.06534654478471713218347486185, −10.31635335272397056064590550797, −10.26979916749910016244597506302, −9.656478578336960530803698753126, −9.152193956593355017179658185189, −8.870280280708048604130048125561, −8.582199632440406613455595974165, −7.82511348005378655248352941424, −6.95568505175323002066440911478, −6.81453495192950695903546593208, −6.63902207319802501609197302043, −5.87257436771050111079399301616, −5.76300241416461614138720331550, −5.09063330831019722904981543595, −4.03437824152938906022421265140, −3.91536476538112132226318090407, −3.13157565604207640057341576461, −2.11314492646761721516340161378, −2.03285933843507897008001771059, −1.13684157457683013998255757108,
1.13684157457683013998255757108, 2.03285933843507897008001771059, 2.11314492646761721516340161378, 3.13157565604207640057341576461, 3.91536476538112132226318090407, 4.03437824152938906022421265140, 5.09063330831019722904981543595, 5.76300241416461614138720331550, 5.87257436771050111079399301616, 6.63902207319802501609197302043, 6.81453495192950695903546593208, 6.95568505175323002066440911478, 7.82511348005378655248352941424, 8.582199632440406613455595974165, 8.870280280708048604130048125561, 9.152193956593355017179658185189, 9.656478578336960530803698753126, 10.26979916749910016244597506302, 10.31635335272397056064590550797, 11.06534654478471713218347486185
