L(s) = 1 | + 2·3-s + 2·9-s + 2·13-s + 4·17-s − 2·19-s + 4·23-s + 6·27-s + 4·29-s + 12·31-s − 14·37-s + 4·39-s + 16·41-s + 8·43-s + 16·47-s + 6·49-s + 8·51-s − 18·53-s − 4·57-s + 12·59-s − 2·67-s + 8·69-s + 12·71-s − 4·73-s + 4·79-s + 11·81-s − 12·83-s + 8·87-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 2/3·9-s + 0.554·13-s + 0.970·17-s − 0.458·19-s + 0.834·23-s + 1.15·27-s + 0.742·29-s + 2.15·31-s − 2.30·37-s + 0.640·39-s + 2.49·41-s + 1.21·43-s + 2.33·47-s + 6/7·49-s + 1.12·51-s − 2.47·53-s − 0.529·57-s + 1.56·59-s − 0.244·67-s + 0.963·69-s + 1.42·71-s − 0.468·73-s + 0.450·79-s + 11/9·81-s − 1.31·83-s + 0.857·87-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
\(L(1)\) |
\(\approx\) |
\(7.626203813\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.626203813\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.966114504457539159640616706764, −7.87506370992720182169816730802, −7.30276772745260410694179205449, −7.22628764223075369461301228505, −6.70506907750284824007959888164, −6.30478645087209414468280431735, −6.05406248128657130735098826405, −5.67239272690558520293588017501, −5.18578726361425084528100652820, −4.75560656792033809391618071536, −4.50765887367555729721685928172, −4.04492494121050964872610391564, −3.61785217134958623400006565115, −3.37066685080786503067386699734, −2.71440030091403451111671447468, −2.65503796651228161044879027896, −2.24097289330052058671662929389, −1.52571952422545531473488459299, −0.852171786318819812294434282395, −0.836773973258976486139231782451,
0.836773973258976486139231782451, 0.852171786318819812294434282395, 1.52571952422545531473488459299, 2.24097289330052058671662929389, 2.65503796651228161044879027896, 2.71440030091403451111671447468, 3.37066685080786503067386699734, 3.61785217134958623400006565115, 4.04492494121050964872610391564, 4.50765887367555729721685928172, 4.75560656792033809391618071536, 5.18578726361425084528100652820, 5.67239272690558520293588017501, 6.05406248128657130735098826405, 6.30478645087209414468280431735, 6.70506907750284824007959888164, 7.22628764223075369461301228505, 7.30276772745260410694179205449, 7.87506370992720182169816730802, 7.966114504457539159640616706764