| L(s) = 1 | + 3·3-s + 2·5-s + 3·7-s + 4·9-s − 3·11-s + 13-s + 6·15-s + 11·17-s − 5·19-s + 9·21-s − 2·23-s + 3·25-s + 6·27-s − 10·29-s + 5·31-s − 9·33-s + 6·35-s + 4·37-s + 3·39-s − 13·41-s + 8·43-s + 8·45-s + 14·47-s − 4·49-s + 33·51-s − 6·55-s − 15·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s + 1.13·7-s + 4/3·9-s − 0.904·11-s + 0.277·13-s + 1.54·15-s + 2.66·17-s − 1.14·19-s + 1.96·21-s − 0.417·23-s + 3/5·25-s + 1.15·27-s − 1.85·29-s + 0.898·31-s − 1.56·33-s + 1.01·35-s + 0.657·37-s + 0.480·39-s − 2.03·41-s + 1.21·43-s + 1.19·45-s + 2.04·47-s − 4/7·49-s + 4.62·51-s − 0.809·55-s − 1.98·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.85261541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.85261541\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 61 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 65 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13 T + 121 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 119 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 151 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 226 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T - 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 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
−8.112662492387008552132388305765, −8.002912225095064904277172679203, −7.43370896929000770674778093950, −7.31305912939638893808623971475, −6.80900060976374921837268552928, −6.29333969318942529811688635287, −5.86919248505256673419476323201, −5.67421924558731445474856511183, −5.12437900245298959419213104552, −5.06345729238177616434439874867, −4.52333517359616098617612015751, −3.94747818359360939156910204163, −3.57643016570454999220952286383, −3.41695131955969658481557877061, −2.78932159777582304299458928396, −2.47927197662107214792109232126, −1.97847992146891655027362243393, −1.92251407453233566532867499728, −1.13346635341198203923847893046, −0.73150734587939924496508066816,
0.73150734587939924496508066816, 1.13346635341198203923847893046, 1.92251407453233566532867499728, 1.97847992146891655027362243393, 2.47927197662107214792109232126, 2.78932159777582304299458928396, 3.41695131955969658481557877061, 3.57643016570454999220952286383, 3.94747818359360939156910204163, 4.52333517359616098617612015751, 5.06345729238177616434439874867, 5.12437900245298959419213104552, 5.67421924558731445474856511183, 5.86919248505256673419476323201, 6.29333969318942529811688635287, 6.80900060976374921837268552928, 7.31305912939638893808623971475, 7.43370896929000770674778093950, 8.002912225095064904277172679203, 8.112662492387008552132388305765
