L(s) = 1 | + (−1.41 + 1.41i)7-s + i·9-s + (−1.41 − 1.41i)23-s − 2·41-s + (−1.41 + 1.41i)47-s − 3.00i·49-s + (−1.41 − 1.41i)63-s − 81-s + 2i·89-s + (1.41 + 1.41i)103-s + ⋯ |
L(s) = 1 | + (−1.41 + 1.41i)7-s + i·9-s + (−1.41 − 1.41i)23-s − 2·41-s + (−1.41 + 1.41i)47-s − 3.00i·49-s + (−1.41 − 1.41i)63-s − 81-s + 2i·89-s + (1.41 + 1.41i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4347297935\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4347297935\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + 2T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - 2iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.148326532307363632626620187203, −8.456745219485019832428474431609, −7.893583112563146795830117800031, −6.69503468329614598464856202422, −6.26241272749764046438391669309, −5.46083967693233716720656857596, −4.69652154913408887139214636706, −3.54382545189060467677852852507, −2.67783678238172333166591511944, −1.99642760280359098530146809054,
0.24654595920807105591038488511, 1.62408912239523120760558423085, 3.28993053408557069037493646195, 3.55069198096291171232947161275, 4.40442741252701725036055526623, 5.63017924837728775204799129432, 6.41473516732947081626218693252, 6.91657645981859781069903239951, 7.59710677473257601591282565970, 8.533938438053315476957157777162