| L(s) = 1 | − 4-s − 9-s + 8·11-s + 16-s + 8·19-s − 12·29-s − 16·31-s + 36-s + 4·41-s − 8·44-s − 49-s − 8·59-s − 4·61-s − 64-s + 16·71-s − 8·76-s + 81-s − 4·89-s − 8·99-s + 12·101-s + 4·109-s + 12·116-s + 26·121-s + 16·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s + 1.83·19-s − 2.22·29-s − 2.87·31-s + 1/6·36-s + 0.624·41-s − 1.20·44-s − 1/7·49-s − 1.04·59-s − 0.512·61-s − 1/8·64-s + 1.89·71-s − 0.917·76-s + 1/9·81-s − 0.423·89-s − 0.804·99-s + 1.19·101-s + 0.383·109-s + 1.11·116-s + 2.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.910543939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910543939\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.859652272425350511902305509961, −9.447057283569130441417424344514, −9.370208480387622827888696954544, −8.983620372937779108825966607753, −8.785176331813305272340846746344, −7.87681988433385233581362633555, −7.69816816059165889946867072630, −7.13419094354333469621204911255, −6.92816056742069842746969265905, −6.18196090431477404226308983334, −5.90029835048896365594174538082, −5.28987120082654965972555168831, −5.15580479243435761895332316844, −4.09468720214872204875815967242, −4.03658464919570634281663503432, −3.43276048838500368094917765351, −3.12335468281836056047703891757, −1.86731599237915474104884241327, −1.62691749834673692528425413821, −0.65970352069144420730694857823,
0.65970352069144420730694857823, 1.62691749834673692528425413821, 1.86731599237915474104884241327, 3.12335468281836056047703891757, 3.43276048838500368094917765351, 4.03658464919570634281663503432, 4.09468720214872204875815967242, 5.15580479243435761895332316844, 5.28987120082654965972555168831, 5.90029835048896365594174538082, 6.18196090431477404226308983334, 6.92816056742069842746969265905, 7.13419094354333469621204911255, 7.69816816059165889946867072630, 7.87681988433385233581362633555, 8.785176331813305272340846746344, 8.983620372937779108825966607753, 9.370208480387622827888696954544, 9.447057283569130441417424344514, 9.859652272425350511902305509961
