| L(s) = 1 | + 2-s − 5-s + 7-s − 8-s − 10-s + 6·11-s − 2·13-s + 14-s − 16-s − 8·19-s + 6·22-s + 9·23-s − 2·26-s + 3·29-s + 4·31-s − 35-s + 16·37-s − 8·38-s + 40-s − 3·41-s − 8·43-s + 9·46-s − 3·47-s + 7·49-s − 12·53-s − 6·55-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.83·19-s + 1.27·22-s + 1.87·23-s − 0.392·26-s + 0.557·29-s + 0.718·31-s − 0.169·35-s + 2.63·37-s − 1.29·38-s + 0.158·40-s − 0.468·41-s − 1.21·43-s + 1.32·46-s − 0.437·47-s + 49-s − 1.64·53-s − 0.809·55-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.011191386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011191386\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−12.29565176751956594389221883950, −11.45825519579322713669543358880, −11.44317022022064513711520159055, −11.10327927746704489765750144472, −10.21975603427684768872837238495, −9.831429725717679783161143590908, −9.197717839934871945562510310812, −8.866087511776889307237491098761, −8.228898260932424032816659437545, −7.951352423548577127216843386143, −6.97594127203070825942260421728, −6.50860808150010713448112398084, −6.48286754821777669967409067045, −5.40350590323350401534862511154, −4.89971133213978337552242591701, −4.16708828367489689574721666012, −4.09270346113979369303081087474, −3.09891448510128214338262702386, −2.33452941350093903212636563075, −1.10354538242389848236859212257,
1.10354538242389848236859212257, 2.33452941350093903212636563075, 3.09891448510128214338262702386, 4.09270346113979369303081087474, 4.16708828367489689574721666012, 4.89971133213978337552242591701, 5.40350590323350401534862511154, 6.48286754821777669967409067045, 6.50860808150010713448112398084, 6.97594127203070825942260421728, 7.951352423548577127216843386143, 8.228898260932424032816659437545, 8.866087511776889307237491098761, 9.197717839934871945562510310812, 9.831429725717679783161143590908, 10.21975603427684768872837238495, 11.10327927746704489765750144472, 11.44317022022064513711520159055, 11.45825519579322713669543358880, 12.29565176751956594389221883950
