L(s) = 1 | + (−1 − i)2-s − 2·3-s + 2i·4-s + (−2 − i)5-s + (2 + 2i)6-s + (−3 + 3i)7-s + (2 − 2i)8-s + 9-s + (1 + 3i)10-s + (−1 − i)11-s − 4i·12-s − 2i·13-s + 6·14-s + (4 + 2i)15-s − 4·16-s + (1 − i)17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s − 1.15·3-s + i·4-s + (−0.894 − 0.447i)5-s + (0.816 + 0.816i)6-s + (−1.13 + 1.13i)7-s + (0.707 − 0.707i)8-s + 0.333·9-s + (0.316 + 0.948i)10-s + (−0.301 − 0.301i)11-s − 1.15i·12-s − 0.554i·13-s + 1.60·14-s + (1.03 + 0.516i)15-s − 16-s + (0.242 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 5 | \( 1 + (2 + i)T \) |
good | 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 + (1 + i)T + 11iT^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-1 + i)T - 17iT^{2} \) |
| 19 | \( 1 + (3 + 3i)T + 19iT^{2} \) |
| 23 | \( 1 + (1 + i)T + 23iT^{2} \) |
| 29 | \( 1 + (7 - 7i)T - 29iT^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (7 + 7i)T + 47iT^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (1 + i)T + 61iT^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3 + 3i)T - 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (11 - 11i)T - 97iT^{2} \) |
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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
−12.96317739991569446309415491808, −12.43225073815099575731022978335, −11.56026651288003495969755770482, −10.68161814271764683129864823345, −9.307682822426627168383887243707, −8.294765114665058736602328411165, −6.69179242865704678912310909522, −5.18013618333723223342449259876, −3.19744395278082895586299779894, 0,
4.16929381792922326649904005931, 5.99255617960231670738492661515, 6.88805760583972888980449229505, 7.88638268394394799902523829232, 9.699859526262278865980790027689, 10.61649948138300935650475551650, 11.43204301764673511085392549231, 12.80830581494530963976800378598, 14.21442938078161757019809759424