L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 2·9-s + 2·11-s + 4·12-s + 2·13-s − 4·16-s + 4·17-s + 4·18-s + 6·19-s + 4·22-s + 4·26-s + 6·27-s + 6·29-s − 16·31-s − 8·32-s + 4·33-s + 8·34-s + 4·36-s − 6·37-s + 12·38-s + 4·39-s − 10·43-s + 4·44-s − 16·47-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 2/3·9-s + 0.603·11-s + 1.15·12-s + 0.554·13-s − 16-s + 0.970·17-s + 0.942·18-s + 1.37·19-s + 0.852·22-s + 0.784·26-s + 1.15·27-s + 1.11·29-s − 2.87·31-s − 1.41·32-s + 0.696·33-s + 1.37·34-s + 2/3·36-s − 0.986·37-s + 1.94·38-s + 0.640·39-s − 1.52·43-s + 0.603·44-s − 2.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.368365887\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.368365887\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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
−11.84911762097999129707298291672, −11.17995682663303422055461565887, −10.67070376332322671429443172713, −10.13410004926732766363430857796, −9.608631465290363926809972280931, −9.189908910893518952998623897362, −8.640597872050038630030658229228, −8.488333513118573526587241989754, −7.62330064532435654482668056130, −7.26709932159447023305436676189, −6.80529293399016414540093640608, −6.12667229047235583286789158836, −5.69888111454043119849352984857, −4.97244842063249222586446373274, −4.75571725058166857374075900092, −3.72481266677675034918624972780, −3.32789753645640897773335918179, −3.29813004902722647931472395788, −2.21818377024386315730845355357, −1.40046900725607102554863375223,
1.40046900725607102554863375223, 2.21818377024386315730845355357, 3.29813004902722647931472395788, 3.32789753645640897773335918179, 3.72481266677675034918624972780, 4.75571725058166857374075900092, 4.97244842063249222586446373274, 5.69888111454043119849352984857, 6.12667229047235583286789158836, 6.80529293399016414540093640608, 7.26709932159447023305436676189, 7.62330064532435654482668056130, 8.488333513118573526587241989754, 8.640597872050038630030658229228, 9.189908910893518952998623897362, 9.608631465290363926809972280931, 10.13410004926732766363430857796, 10.67070376332322671429443172713, 11.17995682663303422055461565887, 11.84911762097999129707298291672