| L(s) = 1 | − 2·3-s + 2·5-s + 6·7-s + 3·9-s − 2·11-s − 4·15-s + 6·17-s − 6·19-s − 12·21-s − 2·23-s − 25-s − 4·27-s + 6·29-s + 4·33-s + 12·35-s + 6·45-s − 10·47-s + 18·49-s − 12·51-s + 12·53-s − 4·55-s + 12·57-s − 14·59-s + 18·61-s + 18·63-s + 4·69-s − 16·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 2.26·7-s + 9-s − 0.603·11-s − 1.03·15-s + 1.45·17-s − 1.37·19-s − 2.61·21-s − 0.417·23-s − 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.696·33-s + 2.02·35-s + 0.894·45-s − 1.45·47-s + 18/7·49-s − 1.68·51-s + 1.64·53-s − 0.539·55-s + 1.58·57-s − 1.82·59-s + 2.30·61-s + 2.26·63-s + 0.481·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.497171632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.497171632\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.600659163403943340721147380060, −8.732416131695606304152432953861, −8.660084148817916152346753387188, −8.179671446699195864315195487385, −7.88531276364715176598974590460, −7.44831913843281702122816080154, −7.07585026775653029031681084915, −6.52360950633621555241054433595, −6.05480847055099857453610469966, −5.70610726204098286231775101142, −5.46918863896066659462475474015, −4.92874639674608726439219222042, −4.70748636658189433546918952988, −4.30161627428678488144500822525, −3.73861864982597786078589367606, −2.93447425977994265750173322873, −2.26606974793619568862684863491, −1.77856143244691532109586055712, −1.42982581633902156843504497113, −0.64800438171117408944877482685,
0.64800438171117408944877482685, 1.42982581633902156843504497113, 1.77856143244691532109586055712, 2.26606974793619568862684863491, 2.93447425977994265750173322873, 3.73861864982597786078589367606, 4.30161627428678488144500822525, 4.70748636658189433546918952988, 4.92874639674608726439219222042, 5.46918863896066659462475474015, 5.70610726204098286231775101142, 6.05480847055099857453610469966, 6.52360950633621555241054433595, 7.07585026775653029031681084915, 7.44831913843281702122816080154, 7.88531276364715176598974590460, 8.179671446699195864315195487385, 8.660084148817916152346753387188, 8.732416131695606304152432953861, 9.600659163403943340721147380060
