L(s) = 1 | + 2·3-s − 4-s + 2·5-s + 9-s − 2·12-s + 13-s + 4·15-s − 17-s + 19-s − 2·20-s − 23-s + 3·25-s − 2·27-s + 2·31-s − 36-s − 37-s + 2·39-s + 41-s − 43-s + 2·45-s − 49-s − 2·51-s − 52-s + 2·53-s + 2·57-s + 59-s − 4·60-s + ⋯ |
L(s) = 1 | + 2·3-s − 4-s + 2·5-s + 9-s − 2·12-s + 13-s + 4·15-s − 17-s + 19-s − 2·20-s − 23-s + 3·25-s − 2·27-s + 2·31-s − 36-s − 37-s + 2·39-s + 41-s − 43-s + 2·45-s − 49-s − 2·51-s − 52-s + 2·53-s + 2·57-s + 59-s − 4·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4060225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4060225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.712141317\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.712141317\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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
−9.564342273201083170771894671525, −8.973437830696771282244825353059, −8.835694282105224245604335065364, −8.446012274705330453190676880844, −8.433425281356616107840097949216, −7.73077641416160844032217067853, −7.38635798894064934213762770605, −6.67191979253310352585543787114, −6.33630492288886064595061644760, −6.07674539639486189808516648918, −5.36473189576526764261045781321, −5.27118353993412501623955610518, −4.66470846941597576815250329669, −3.94787282790723248587260014878, −3.83518184903051328584385722872, −3.03126444200120827247587212754, −2.73804189827363699147307946678, −2.33761764184055697666747542262, −1.82371456114366736857404340108, −1.18305123503357502726197551060,
1.18305123503357502726197551060, 1.82371456114366736857404340108, 2.33761764184055697666747542262, 2.73804189827363699147307946678, 3.03126444200120827247587212754, 3.83518184903051328584385722872, 3.94787282790723248587260014878, 4.66470846941597576815250329669, 5.27118353993412501623955610518, 5.36473189576526764261045781321, 6.07674539639486189808516648918, 6.33630492288886064595061644760, 6.67191979253310352585543787114, 7.38635798894064934213762770605, 7.73077641416160844032217067853, 8.433425281356616107840097949216, 8.446012274705330453190676880844, 8.835694282105224245604335065364, 8.973437830696771282244825353059, 9.564342273201083170771894671525