| L(s) = 1 | − 2-s + 3-s − 6-s + 5·7-s + 8-s + 2·11-s + 8·13-s − 5·14-s − 16-s − 5·17-s + 4·19-s + 5·21-s − 2·22-s + 5·23-s + 24-s − 8·26-s − 27-s − 12·29-s + 11·31-s + 2·33-s + 5·34-s + 8·37-s − 4·38-s + 8·39-s + 10·41-s − 5·42-s − 5·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s + 1.88·7-s + 0.353·8-s + 0.603·11-s + 2.21·13-s − 1.33·14-s − 1/4·16-s − 1.21·17-s + 0.917·19-s + 1.09·21-s − 0.426·22-s + 1.04·23-s + 0.204·24-s − 1.56·26-s − 0.192·27-s − 2.22·29-s + 1.97·31-s + 0.348·33-s + 0.857·34-s + 1.31·37-s − 0.648·38-s + 1.28·39-s + 1.56·41-s − 0.771·42-s − 0.737·46-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
| \(L(1)\) |
\(\approx\) |
\(2.921898080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.921898080\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 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
−9.866432794298901754021293097687, −9.701070766085059643832054411710, −8.950798334090935621630052386204, −8.883064094953710124170308552058, −8.614171111667303945006922285519, −8.190847498103796296484384017440, −7.70128489669497258015131771057, −7.46866954005042413085886940372, −6.93254852802980463489676724458, −6.36365353699598227377158881534, −5.72578682049746205907331564486, −5.61356123155920467164289765234, −4.76534720584507858946737659430, −4.22928170438836091623492441856, −4.14838517495850599101002565280, −3.38930087982943395120285233898, −2.65851485580606563153768785084, −2.05653696486711078727303000129, −1.18800209187958981410913311556, −1.13310188014469602296077227228,
1.13310188014469602296077227228, 1.18800209187958981410913311556, 2.05653696486711078727303000129, 2.65851485580606563153768785084, 3.38930087982943395120285233898, 4.14838517495850599101002565280, 4.22928170438836091623492441856, 4.76534720584507858946737659430, 5.61356123155920467164289765234, 5.72578682049746205907331564486, 6.36365353699598227377158881534, 6.93254852802980463489676724458, 7.46866954005042413085886940372, 7.70128489669497258015131771057, 8.190847498103796296484384017440, 8.614171111667303945006922285519, 8.883064094953710124170308552058, 8.950798334090935621630052386204, 9.701070766085059643832054411710, 9.866432794298901754021293097687
