L(s) = 1 | + 2-s + 4-s + 8-s + 11-s + 16-s − 17-s − 19-s + 22-s + 32-s − 34-s − 38-s + 41-s − 2·43-s + 44-s + 49-s − 2·59-s + 64-s + 67-s − 68-s + 73-s − 76-s + 82-s − 83-s − 2·86-s + 88-s + 89-s − 2·97-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 8-s + 11-s + 16-s − 17-s − 19-s + 22-s + 32-s − 34-s − 38-s + 41-s − 2·43-s + 44-s + 49-s − 2·59-s + 64-s + 67-s − 68-s + 73-s − 76-s + 82-s − 83-s − 2·86-s + 88-s + 89-s − 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.202820647\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.202820647\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.465625098131355918188403281033, −8.642810312287221008585754063677, −7.75047902300715182823950980305, −6.67833116003745239717115701629, −6.42242258273585235599389438989, −5.33686562578059969849304683945, −4.40719362115053941726705538025, −3.81255898184309720005267252915, −2.65353599889505161743456569945, −1.62572052382060186085981353731,
1.62572052382060186085981353731, 2.65353599889505161743456569945, 3.81255898184309720005267252915, 4.40719362115053941726705538025, 5.33686562578059969849304683945, 6.42242258273585235599389438989, 6.67833116003745239717115701629, 7.75047902300715182823950980305, 8.642810312287221008585754063677, 9.465625098131355918188403281033