L(s) = 1 | + 2·3-s + 3·9-s − 4·11-s + 4·13-s − 8·17-s + 4·23-s + 2·25-s + 4·27-s − 4·29-s + 8·31-s − 8·33-s + 4·37-s + 8·39-s − 16·41-s + 16·43-s − 8·47-s − 16·51-s + 4·53-s − 16·59-s − 4·61-s − 8·67-s + 8·69-s + 12·71-s − 12·73-s + 4·75-s − 8·79-s + 5·81-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 1.20·11-s + 1.10·13-s − 1.94·17-s + 0.834·23-s + 2/5·25-s + 0.769·27-s − 0.742·29-s + 1.43·31-s − 1.39·33-s + 0.657·37-s + 1.28·39-s − 2.49·41-s + 2.43·43-s − 1.16·47-s − 2.24·51-s + 0.549·53-s − 2.08·59-s − 0.512·61-s − 0.977·67-s + 0.963·69-s + 1.42·71-s − 1.40·73-s + 0.461·75-s − 0.900·79-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.671070684\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.671070684\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.81752263130246188765490064504, −7.65651325732566230463417525634, −7.29287451208356264095984422391, −6.81465441514980751787628056285, −6.53337898913859922368373281997, −6.41475120588702610241165444292, −5.70223318803946767535037767441, −5.59189357643917127795823790665, −5.00919488270935039219713240695, −4.61628337285320741375164272825, −4.25312523685626085477505613103, −4.23773742306480475982478718798, −3.46540363544261841324593555692, −3.16220555108758974624089982498, −2.70174786262735373283221458002, −2.68514697708499295060152238655, −1.93407079451544467341685431602, −1.65944002383093194328141439287, −1.10128092783868310464318061590, −0.33798999570611027605643578164,
0.33798999570611027605643578164, 1.10128092783868310464318061590, 1.65944002383093194328141439287, 1.93407079451544467341685431602, 2.68514697708499295060152238655, 2.70174786262735373283221458002, 3.16220555108758974624089982498, 3.46540363544261841324593555692, 4.23773742306480475982478718798, 4.25312523685626085477505613103, 4.61628337285320741375164272825, 5.00919488270935039219713240695, 5.59189357643917127795823790665, 5.70223318803946767535037767441, 6.41475120588702610241165444292, 6.53337898913859922368373281997, 6.81465441514980751787628056285, 7.29287451208356264095984422391, 7.65651325732566230463417525634, 7.81752263130246188765490064504