L(s) = 1 | + (0.385 − 0.222i)2-s + (−0.400 + 0.694i)4-s + (−0.866 − 0.5i)7-s + 0.801i·8-s + (−0.5 + 0.866i)9-s + (−1.56 + 0.900i)11-s − 0.445·14-s + (−0.222 − 0.385i)16-s + 0.445i·18-s + (−0.400 + 0.694i)22-s + (0.623 + 1.07i)23-s − 25-s + (0.694 − 0.400i)28-s + (0.222 + 0.385i)29-s + (−0.866 − 0.5i)32-s + ⋯ |
L(s) = 1 | + (0.385 − 0.222i)2-s + (−0.400 + 0.694i)4-s + (−0.866 − 0.5i)7-s + 0.801i·8-s + (−0.5 + 0.866i)9-s + (−1.56 + 0.900i)11-s − 0.445·14-s + (−0.222 − 0.385i)16-s + 0.445i·18-s + (−0.400 + 0.694i)22-s + (0.623 + 1.07i)23-s − 25-s + (0.694 − 0.400i)28-s + (0.222 + 0.385i)29-s + (−0.866 − 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6286429199\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6286429199\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.385 + 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−10.23242928589298149343564125924, −9.530877922004027921744150123099, −8.510619940444083116350698702591, −7.62259513965049867799151388154, −7.29214038976976874643878621614, −5.77245720157717041110395799796, −5.04925188764025914687450419383, −4.15291728925564051343313597353, −3.08124370031055980767532441767, −2.31773025899212567698055362782,
0.46246844166766787470249264948, 2.59326815834271383735362469095, 3.45770188230477458205202466461, 4.64829770245183925765025959500, 5.74059673786470362816093837827, 5.99041009311565771818719593870, 6.95277257310320788845356982190, 8.233936241205362841505167445621, 8.892930608049712692590132223203, 9.738260222229850093450566364253