| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 2·7-s − 2·8-s − 2·10-s − 12-s − 2·14-s − 2·15-s + 2·16-s − 17-s + 2·20-s − 2·21-s − 23-s + 2·24-s + 25-s + 27-s + 2·28-s + 2·29-s + 2·30-s − 2·32-s + 34-s + 4·35-s − 37-s − 4·40-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 2·7-s − 2·8-s − 2·10-s − 12-s − 2·14-s − 2·15-s + 2·16-s − 17-s + 2·20-s − 2·21-s − 23-s + 2·24-s + 25-s + 27-s + 2·28-s + 2·29-s + 2·30-s − 2·32-s + 34-s + 4·35-s − 37-s − 4·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6437457594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6437457594\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−10.41020402413973370155607879926, −10.24466712047447992575736127642, −10.24308271844872838731213624810, −9.289819941847058277306485181826, −9.187723038673504401665313091630, −8.546389109263076573488769454453, −8.344418111701164508359424415217, −7.985276601261947577574629006785, −7.06640255600404315903579374738, −6.82137228108681970668450077020, −6.17174219760518265269628121059, −6.04287569918406175931976095801, −5.37402667149076234526353148970, −5.35508054284277644058245191365, −4.63556605716742529209449460992, −3.92520549731445226877370522291, −2.78277836838499132706963186753, −2.33037037427067170914213894285, −1.88287761886260851818065718850, −1.15644882459045300467732193160,
1.15644882459045300467732193160, 1.88287761886260851818065718850, 2.33037037427067170914213894285, 2.78277836838499132706963186753, 3.92520549731445226877370522291, 4.63556605716742529209449460992, 5.35508054284277644058245191365, 5.37402667149076234526353148970, 6.04287569918406175931976095801, 6.17174219760518265269628121059, 6.82137228108681970668450077020, 7.06640255600404315903579374738, 7.985276601261947577574629006785, 8.344418111701164508359424415217, 8.546389109263076573488769454453, 9.187723038673504401665313091630, 9.289819941847058277306485181826, 10.24308271844872838731213624810, 10.24466712047447992575736127642, 10.41020402413973370155607879926
