L(s) = 1 | − 3·2-s + 2·3-s + 4·4-s − 6·6-s − 3·8-s + 3·9-s + 8·12-s + 5·13-s + 3·16-s − 3·17-s − 9·18-s + 6·19-s + 6·23-s − 6·24-s + 7·25-s − 15·26-s + 10·27-s − 3·29-s − 6·32-s + 9·34-s + 12·36-s + 15·37-s − 18·38-s + 10·39-s + 9·41-s − 8·43-s − 18·46-s + ⋯ |
L(s) = 1 | − 2.12·2-s + 1.15·3-s + 2·4-s − 2.44·6-s − 1.06·8-s + 9-s + 2.30·12-s + 1.38·13-s + 3/4·16-s − 0.727·17-s − 2.12·18-s + 1.37·19-s + 1.25·23-s − 1.22·24-s + 7/5·25-s − 2.94·26-s + 1.92·27-s − 0.557·29-s − 1.06·32-s + 1.54·34-s + 2·36-s + 2.46·37-s − 2.91·38-s + 1.60·39-s + 1.40·41-s − 1.21·43-s − 2.65·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.129235079\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.129235079\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 15 T + 112 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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
−10.45077125195733125310922020347, −10.35634903222647307214249563420, −9.523042953568066935061081759641, −9.368023433848938627261473378530, −9.093346863751488825405663314713, −8.784606032870381705088067350927, −8.312282453542354675124519549229, −7.87655938772660249829676498002, −7.67708443227877586900418878810, −7.12105337510844211703819790809, −6.42761798679709011725480848724, −6.32759656866695939344260748572, −5.31126613198605125588421274489, −4.78070743596815260235133973597, −4.12940735671574564333337824063, −3.28612597031454256910109698669, −3.03066183000832866525146587183, −2.27433639748080715438872033636, −1.12344547484835562828742641585, −1.05120114925961720351497648493,
1.05120114925961720351497648493, 1.12344547484835562828742641585, 2.27433639748080715438872033636, 3.03066183000832866525146587183, 3.28612597031454256910109698669, 4.12940735671574564333337824063, 4.78070743596815260235133973597, 5.31126613198605125588421274489, 6.32759656866695939344260748572, 6.42761798679709011725480848724, 7.12105337510844211703819790809, 7.67708443227877586900418878810, 7.87655938772660249829676498002, 8.312282453542354675124519549229, 8.784606032870381705088067350927, 9.093346863751488825405663314713, 9.368023433848938627261473378530, 9.523042953568066935061081759641, 10.35634903222647307214249563420, 10.45077125195733125310922020347