L(s) = 1 | − 2-s + 4-s + (−2.62 − 0.358i)7-s − 8-s + (−2.12 + 3.67i)11-s + (1.12 − 1.94i)13-s + (2.62 + 0.358i)14-s + 16-s + (1.12 − 1.94i)19-s + (2.12 − 3.67i)22-s + (−0.621 − 1.07i)23-s + (2.5 − 4.33i)25-s + (−1.12 + 1.94i)26-s + (−2.62 − 0.358i)28-s + (2.12 + 3.67i)29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.990 − 0.135i)7-s − 0.353·8-s + (−0.639 + 1.10i)11-s + (0.310 − 0.538i)13-s + (0.700 + 0.0958i)14-s + 0.250·16-s + (0.257 − 0.445i)19-s + (0.452 − 0.783i)22-s + (−0.129 − 0.224i)23-s + (0.5 − 0.866i)25-s + (−0.219 + 0.380i)26-s + (−0.495 − 0.0677i)28-s + (0.393 + 0.682i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8424291597\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8424291597\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.62 + 0.358i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.12 + 1.94i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 + 1.94i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.621 + 1.07i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.12 - 3.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.74 + 9.94i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.24 + 9.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 + (2.12 + 3.67i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 0.242T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 0.757T + 79T^{2} \) |
| 83 | \( 1 + (-8.12 - 14.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.74 + 9.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.24 + 3.88i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.860467720890095960511629410799, −8.898105893728767224521319820606, −8.134198175471581296168146362528, −7.17196414325511648224557787967, −6.61812369212987268651370680819, −5.56980416333976354319365072486, −4.46787822466665036350590052947, −3.18599299319625536952875040931, −2.27184772789444736902443874373, −0.56026880286845645292648662285,
1.05829319207624412608838134597, 2.71444585009447647173835229913, 3.42096954632153069693973601571, 4.83470861529287452892331775937, 6.18742400007649537410871618497, 6.36585166992815776476110421962, 7.71141285746211010885598933677, 8.283079058547677689581959094703, 9.230488728019413895210086117473, 9.810972429363899442158760222063