| L(s) = 1 | − 8·5-s − 24·13-s − 30·17-s + 39·25-s − 40·29-s − 24·37-s + 18·41-s + 49·49-s + 56·53-s − 120·61-s + 192·65-s − 110·73-s + 240·85-s + 78·89-s − 130·97-s − 40·101-s − 120·109-s + 30·113-s + ⋯ |
| L(s) = 1 | − 8/5·5-s − 1.84·13-s − 1.76·17-s + 1.55·25-s − 1.37·29-s − 0.648·37-s + 0.439·41-s + 49-s + 1.05·53-s − 1.96·61-s + 2.95·65-s − 1.50·73-s + 2.82·85-s + 0.876·89-s − 1.34·97-s − 0.396·101-s − 1.10·109-s + 0.265·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3926445416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3926445416\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 24 T + p^{2} T^{2} \) |
| 17 | \( 1 + 30 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 40 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 24 T + p^{2} T^{2} \) |
| 41 | \( 1 - 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 56 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 120 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 78 T + p^{2} T^{2} \) |
| 97 | \( 1 + 130 T + p^{2} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.824335251599564993706104007778, −7.920350736865893228011348792093, −7.29149491981388255029797153180, −6.86209988657697914036118356295, −5.56094657495538527446686215409, −4.55740357393952230607222872959, −4.16499162898086152640489050329, −3.05050329516286356472001076116, −2.08542523457136947841298086095, −0.30467051728936489548594849349,
0.30467051728936489548594849349, 2.08542523457136947841298086095, 3.05050329516286356472001076116, 4.16499162898086152640489050329, 4.55740357393952230607222872959, 5.56094657495538527446686215409, 6.86209988657697914036118356295, 7.29149491981388255029797153180, 7.920350736865893228011348792093, 8.824335251599564993706104007778
