| L(s) = 1 | − 2·2-s + 3·4-s + 4·7-s − 4·8-s − 2·11-s − 8·14-s + 5·16-s − 4·17-s + 2·19-s + 4·22-s + 2·23-s + 12·28-s − 6·29-s − 6·31-s − 6·32-s + 8·34-s + 4·37-s − 4·38-s − 8·41-s − 6·44-s − 4·46-s − 4·47-s + 4·49-s + 10·53-s − 16·56-s + 12·58-s − 8·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.51·7-s − 1.41·8-s − 0.603·11-s − 2.13·14-s + 5/4·16-s − 0.970·17-s + 0.458·19-s + 0.852·22-s + 0.417·23-s + 2.26·28-s − 1.11·29-s − 1.07·31-s − 1.06·32-s + 1.37·34-s + 0.657·37-s − 0.648·38-s − 1.24·41-s − 0.904·44-s − 0.589·46-s − 0.583·47-s + 4/7·49-s + 1.37·53-s − 2.13·56-s + 1.57·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 74 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 125 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 152 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 113 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 203 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 204 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.54069475921091076017814274356, −7.39163410571910708312183102178, −7.17545928201502688642327670180, −6.82622431946099047620903446247, −6.15896956988606658380535165965, −6.09293617390758347831210042054, −5.56854640329403279451188217206, −5.15163857012564456016355525999, −4.89863233991983043494807823760, −4.61179805475251468394872822156, −3.93957184210039296949266937898, −3.68568569855719774062897201111, −3.09867774447564368305507046092, −2.68922285643142487106369187184, −2.15073039618582591710926159104, −1.97215632531114422164980790992, −1.26266098821697939530861211072, −1.23914093564310129754201544855, 0, 0,
1.23914093564310129754201544855, 1.26266098821697939530861211072, 1.97215632531114422164980790992, 2.15073039618582591710926159104, 2.68922285643142487106369187184, 3.09867774447564368305507046092, 3.68568569855719774062897201111, 3.93957184210039296949266937898, 4.61179805475251468394872822156, 4.89863233991983043494807823760, 5.15163857012564456016355525999, 5.56854640329403279451188217206, 6.09293617390758347831210042054, 6.15896956988606658380535165965, 6.82622431946099047620903446247, 7.17545928201502688642327670180, 7.39163410571910708312183102178, 7.54069475921091076017814274356
