| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 8·17-s − 16·19-s + 4·23-s − 4·27-s − 3·36-s + 20·37-s − 2·48-s − 14·49-s − 16·51-s + 32·57-s − 12·59-s − 64-s − 8·68-s − 8·69-s − 4·73-s + 16·76-s + 5·81-s − 28·89-s − 4·92-s − 28·97-s − 32·101-s − 24·107-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 1.94·17-s − 3.67·19-s + 0.834·23-s − 0.769·27-s − 1/2·36-s + 3.28·37-s − 0.288·48-s − 2·49-s − 2.24·51-s + 4.23·57-s − 1.56·59-s − 1/8·64-s − 0.970·68-s − 0.963·69-s − 0.468·73-s + 1.83·76-s + 5/9·81-s − 2.96·89-s − 0.417·92-s − 2.84·97-s − 3.18·101-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3736270966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3736270966\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−9.454455777321155674647064823015, −8.581290318265313772927911058906, −8.251384757920146964513889705113, −7.910673267669454286318488660094, −7.81295900907729808887160602359, −6.85341769607391646679043269417, −6.83948308781699878007791209623, −6.31021889525136825554388906900, −6.02395083310914854289804043141, −5.54573865938881724513349356158, −5.35827001729148619412196653121, −4.55144247813384837498854154590, −4.42188670163783221751846417800, −4.15514062847923329866443643751, −3.56964580108148206526067005639, −2.67544652136829844120163618609, −2.65252113255464495053133799965, −1.39587103380115902539764889328, −1.37888511047699812300227397434, −0.23646081381946053391230287084,
0.23646081381946053391230287084, 1.37888511047699812300227397434, 1.39587103380115902539764889328, 2.65252113255464495053133799965, 2.67544652136829844120163618609, 3.56964580108148206526067005639, 4.15514062847923329866443643751, 4.42188670163783221751846417800, 4.55144247813384837498854154590, 5.35827001729148619412196653121, 5.54573865938881724513349356158, 6.02395083310914854289804043141, 6.31021889525136825554388906900, 6.83948308781699878007791209623, 6.85341769607391646679043269417, 7.81295900907729808887160602359, 7.910673267669454286318488660094, 8.251384757920146964513889705113, 8.581290318265313772927911058906, 9.454455777321155674647064823015
