| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s + 5·13-s + 5·16-s + 3·17-s − 2·18-s + 4·19-s − 25-s − 10·26-s − 6·32-s − 6·34-s + 3·36-s − 8·38-s + 16·43-s + 12·47-s + 5·49-s + 2·50-s + 15·52-s + 9·53-s − 12·59-s + 7·64-s − 8·67-s + 9·68-s − 4·72-s + 12·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1/3·9-s + 1.38·13-s + 5/4·16-s + 0.727·17-s − 0.471·18-s + 0.917·19-s − 1/5·25-s − 1.96·26-s − 1.06·32-s − 1.02·34-s + 1/2·36-s − 1.29·38-s + 2.43·43-s + 1.75·47-s + 5/7·49-s + 0.282·50-s + 2.08·52-s + 1.23·53-s − 1.56·59-s + 7/8·64-s − 0.977·67-s + 1.09·68-s − 0.471·72-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.044733091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.044733091\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 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
−9.110775268594925208237073398626, −8.643611495598430793316803258043, −8.327260399375405407423149103256, −7.61522798925808677683479502810, −7.35417289125071040709504702295, −7.02495662461770894313225111845, −6.17355347051754326221251187587, −5.76463747131475003329342614211, −5.53569610182193646692413434829, −4.35510933906533231645321848065, −3.92283289808617658855123383618, −3.10925825027007912276416249939, −2.53990485392225780945975202621, −1.48885743391755307332868349585, −0.919504219081725949515003496493,
0.919504219081725949515003496493, 1.48885743391755307332868349585, 2.53990485392225780945975202621, 3.10925825027007912276416249939, 3.92283289808617658855123383618, 4.35510933906533231645321848065, 5.53569610182193646692413434829, 5.76463747131475003329342614211, 6.17355347051754326221251187587, 7.02495662461770894313225111845, 7.35417289125071040709504702295, 7.61522798925808677683479502810, 8.327260399375405407423149103256, 8.643611495598430793316803258043, 9.110775268594925208237073398626
