| L(s) = 1 | + 2·3-s − 2·7-s − 9-s + 2·13-s − 2·17-s + 2·19-s − 4·21-s + 2·23-s − 6·27-s + 2·29-s − 4·31-s − 16·37-s + 4·39-s − 4·43-s + 16·47-s − 9·49-s − 4·51-s − 2·53-s + 4·57-s − 26·59-s + 4·61-s + 2·63-s + 2·67-s + 4·69-s + 8·71-s − 22·73-s − 4·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s − 1/3·9-s + 0.554·13-s − 0.485·17-s + 0.458·19-s − 0.872·21-s + 0.417·23-s − 1.15·27-s + 0.371·29-s − 0.718·31-s − 2.63·37-s + 0.640·39-s − 0.609·43-s + 2.33·47-s − 9/7·49-s − 0.560·51-s − 0.274·53-s + 0.529·57-s − 3.38·59-s + 0.512·61-s + 0.251·63-s + 0.244·67-s + 0.481·69-s + 0.949·71-s − 2.57·73-s − 0.450·79-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 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 130 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 26 T + 285 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 124 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 259 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.58779276166441563743617040580, −7.47274426248749046435273115678, −7.11703506560841947066098778786, −6.68487355840286651474417443577, −6.26997476485352492083799351611, −6.13387912532870205516629540606, −5.52270467367078001612124433667, −5.37376804454491076195966768028, −4.77240864266461464736542228598, −4.56899755447865087834530183579, −3.82116721304273428064685761280, −3.66047544529268203942619882145, −3.25858528138114911871609635283, −3.01133055121667666276951432405, −2.55866084198957870358550130561, −2.19314405489446265771422752616, −1.50258850475201859454052607988, −1.25762343820405512166853930086, 0, 0,
1.25762343820405512166853930086, 1.50258850475201859454052607988, 2.19314405489446265771422752616, 2.55866084198957870358550130561, 3.01133055121667666276951432405, 3.25858528138114911871609635283, 3.66047544529268203942619882145, 3.82116721304273428064685761280, 4.56899755447865087834530183579, 4.77240864266461464736542228598, 5.37376804454491076195966768028, 5.52270467367078001612124433667, 6.13387912532870205516629540606, 6.26997476485352492083799351611, 6.68487355840286651474417443577, 7.11703506560841947066098778786, 7.47274426248749046435273115678, 7.58779276166441563743617040580
