| L(s) = 1 | − 2·3-s + 2·5-s − 4·11-s − 2·13-s − 4·15-s + 8·17-s − 2·19-s + 4·23-s + 3·25-s + 2·27-s + 8·31-s + 8·33-s + 2·37-s + 4·39-s + 8·41-s − 8·43-s + 8·47-s − 14·49-s − 16·51-s + 6·53-s − 8·55-s + 4·57-s + 4·59-s + 20·61-s − 4·65-s − 2·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.20·11-s − 0.554·13-s − 1.03·15-s + 1.94·17-s − 0.458·19-s + 0.834·23-s + 3/5·25-s + 0.384·27-s + 1.43·31-s + 1.39·33-s + 0.328·37-s + 0.640·39-s + 1.24·41-s − 1.21·43-s + 1.16·47-s − 2·49-s − 2.24·51-s + 0.824·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s + 2.56·61-s − 0.496·65-s − 0.244·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.305526050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.305526050\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 168 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.069181190337097198490058876220, −7.989016757283442827782244495505, −7.49245561049059689975223088971, −7.15984503953733026199042122562, −6.65565177181366095152095799586, −6.50864472877248975966058036889, −5.88449958217790189506522001377, −5.79864842048174225544277567356, −5.36980063659905628613049227738, −5.22380192212532715203088631149, −4.76136496544171180091690301047, −4.54883780686391483599950672962, −3.76487143304069908929213538718, −3.43729941510657946948796731233, −2.75930834882753997711101508291, −2.69416474803842986388366058074, −2.11933650144255510014304664064, −1.53257382457574985707899959633, −0.70562099199354715333591481619, −0.64397199377802081986455331939,
0.64397199377802081986455331939, 0.70562099199354715333591481619, 1.53257382457574985707899959633, 2.11933650144255510014304664064, 2.69416474803842986388366058074, 2.75930834882753997711101508291, 3.43729941510657946948796731233, 3.76487143304069908929213538718, 4.54883780686391483599950672962, 4.76136496544171180091690301047, 5.22380192212532715203088631149, 5.36980063659905628613049227738, 5.79864842048174225544277567356, 5.88449958217790189506522001377, 6.50864472877248975966058036889, 6.65565177181366095152095799586, 7.15984503953733026199042122562, 7.49245561049059689975223088971, 7.989016757283442827782244495505, 8.069181190337097198490058876220
