| L(s) = 1 | + 2·3-s − 4·5-s − 2·7-s + 2·9-s + 6·13-s − 8·15-s − 6·17-s + 12·19-s − 4·21-s − 6·23-s + 11·25-s + 6·27-s + 8·35-s + 6·37-s + 12·39-s − 12·41-s + 6·43-s − 8·45-s + 18·47-s + 2·49-s − 12·51-s + 10·53-s + 24·57-s + 20·59-s − 24·61-s − 4·63-s − 24·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 0.755·7-s + 2/3·9-s + 1.66·13-s − 2.06·15-s − 1.45·17-s + 2.75·19-s − 0.872·21-s − 1.25·23-s + 11/5·25-s + 1.15·27-s + 1.35·35-s + 0.986·37-s + 1.92·39-s − 1.87·41-s + 0.914·43-s − 1.19·45-s + 2.62·47-s + 2/7·49-s − 1.68·51-s + 1.37·53-s + 3.17·57-s + 2.60·59-s − 3.07·61-s − 0.503·63-s − 2.97·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.252134334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.252134334\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 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
−9.539994810994042521246872505391, −9.520284011433220289967643089439, −8.873495929082855836628155208453, −8.617914218246992681510748451867, −8.319963941414979635705062489178, −7.954351199074866528575922904573, −7.43613424880558753341772018411, −7.16343907583786005929418723647, −6.76078023017890266240201548927, −6.32522873187797143820731578818, −5.53429960314262907686507309015, −5.38203084575206009875689790453, −4.31392515244849746139924828302, −4.16015594184457010061608941458, −3.81078526827458449596811091803, −3.25006326354692888511777464048, −2.93550550580576916165475114295, −2.39747503569600525383894734055, −1.30235978816778227887700498647, −0.65582297781718012055007100399,
0.65582297781718012055007100399, 1.30235978816778227887700498647, 2.39747503569600525383894734055, 2.93550550580576916165475114295, 3.25006326354692888511777464048, 3.81078526827458449596811091803, 4.16015594184457010061608941458, 4.31392515244849746139924828302, 5.38203084575206009875689790453, 5.53429960314262907686507309015, 6.32522873187797143820731578818, 6.76078023017890266240201548927, 7.16343907583786005929418723647, 7.43613424880558753341772018411, 7.954351199074866528575922904573, 8.319963941414979635705062489178, 8.617914218246992681510748451867, 8.873495929082855836628155208453, 9.520284011433220289967643089439, 9.539994810994042521246872505391
