| L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s − 2·11-s + 4·13-s + 4·15-s − 10·19-s + 4·21-s + 6·23-s + 3·25-s + 2·27-s − 6·29-s − 4·31-s + 4·33-s + 4·35-s − 2·37-s − 8·39-s + 6·41-s − 4·43-s + 3·49-s − 18·53-s + 4·55-s + 20·57-s + 4·61-s − 8·65-s + 8·67-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s − 0.603·11-s + 1.10·13-s + 1.03·15-s − 2.29·19-s + 0.872·21-s + 1.25·23-s + 3/5·25-s + 0.384·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s − 1.28·39-s + 0.937·41-s − 0.609·43-s + 3/7·49-s − 2.47·53-s + 0.539·55-s + 2.64·57-s + 0.512·61-s − 0.992·65-s + 0.977·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4509805569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4509805569\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 180 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T - 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.152866219577461517996839408771, −7.88564588145823208291388660786, −7.52971380063835878836447583583, −7.11166102552308387219708627411, −6.58642387484480798592394274879, −6.54884408557104318721660470282, −6.05801450004200455731709333561, −5.92182817318489812330852451783, −5.35942168899941464169703080407, −5.03997177873655003403914594033, −4.74744442373279420810105582594, −4.16096822463364188991509098151, −3.83952629954115571878934888528, −3.60098234726937077738409185561, −2.92307752081918453401733529785, −2.78787195514632618009735907954, −1.93224681108036158845965655421, −1.61313355181238393416060098770, −0.65418231337783191307546868738, −0.29080531880755536608360242344,
0.29080531880755536608360242344, 0.65418231337783191307546868738, 1.61313355181238393416060098770, 1.93224681108036158845965655421, 2.78787195514632618009735907954, 2.92307752081918453401733529785, 3.60098234726937077738409185561, 3.83952629954115571878934888528, 4.16096822463364188991509098151, 4.74744442373279420810105582594, 5.03997177873655003403914594033, 5.35942168899941464169703080407, 5.92182817318489812330852451783, 6.05801450004200455731709333561, 6.54884408557104318721660470282, 6.58642387484480798592394274879, 7.11166102552308387219708627411, 7.52971380063835878836447583583, 7.88564588145823208291388660786, 8.152866219577461517996839408771
