| L(s) = 1 | − 3-s − 7-s − 9-s − 8·11-s + 13-s − 11·17-s + 2·19-s + 21-s + 3·23-s + 29-s + 2·31-s + 8·33-s + 12·37-s − 39-s + 8·41-s − 14·43-s − 4·47-s − 9·49-s + 11·51-s + 5·53-s − 2·57-s − 59-s + 14·61-s + 63-s − 67-s − 3·69-s − 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 1/3·9-s − 2.41·11-s + 0.277·13-s − 2.66·17-s + 0.458·19-s + 0.218·21-s + 0.625·23-s + 0.185·29-s + 0.359·31-s + 1.39·33-s + 1.97·37-s − 0.160·39-s + 1.24·41-s − 2.13·43-s − 0.583·47-s − 9/7·49-s + 1.54·51-s + 0.686·53-s − 0.264·57-s − 0.130·59-s + 1.79·61-s + 0.125·63-s − 0.122·67-s − 0.361·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 130 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 128 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 198 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.979194033709205134588573104057, −7.21463299875511577164186771431, −6.89044711893253035014160629202, −6.77676453572229197921407598262, −6.25143737185503514656713328076, −6.03020097779723098391741663527, −5.45884980512601390726837987513, −5.37551063084150066520473559135, −4.87303087057931113078154348436, −4.63260863803923493618589361226, −4.24045520236647840013916774093, −3.82128941654395237845097655473, −3.08095165122504694278971159328, −2.92422291702812489703402103132, −2.41339616699632241272366235058, −2.28722451329898414776339778583, −1.53809868931927291377650552111, −0.800606784264878348252739196403, 0, 0,
0.800606784264878348252739196403, 1.53809868931927291377650552111, 2.28722451329898414776339778583, 2.41339616699632241272366235058, 2.92422291702812489703402103132, 3.08095165122504694278971159328, 3.82128941654395237845097655473, 4.24045520236647840013916774093, 4.63260863803923493618589361226, 4.87303087057931113078154348436, 5.37551063084150066520473559135, 5.45884980512601390726837987513, 6.03020097779723098391741663527, 6.25143737185503514656713328076, 6.77676453572229197921407598262, 6.89044711893253035014160629202, 7.21463299875511577164186771431, 7.979194033709205134588573104057
