L(s) = 1 | + (31.9 + 0.0155i)2-s + (−39.7 + 39.7i)3-s + (1.02e3 + 0.998i)4-s + (−4.09e3 + 4.09e3i)5-s + (−1.27e3 + 1.27e3i)6-s − 7.86e3·7-s + (3.27e4 + 47.9i)8-s + 5.58e4i·9-s + (−1.31e5 + 1.31e5i)10-s + (−4.76e4 − 4.76e4i)11-s + (−4.07e4 + 4.06e4i)12-s + (2.25e5 + 2.25e5i)13-s + (−2.51e5 − 122. i)14-s − 3.25e5i·15-s + (1.04e6 + 2.04e3i)16-s − 7.84e5·17-s + ⋯ |
L(s) = 1 | + (0.999 + 0.000487i)2-s + (−0.163 + 0.163i)3-s + (0.999 + 0.000974i)4-s + (−1.31 + 1.31i)5-s + (−0.163 + 0.163i)6-s − 0.468·7-s + (0.999 + 0.00146i)8-s + 0.946i·9-s + (−1.31 + 1.31i)10-s + (−0.295 − 0.295i)11-s + (−0.163 + 0.163i)12-s + (0.606 + 0.606i)13-s + (−0.468 − 0.000228i)14-s − 0.428i·15-s + (0.999 + 0.00194i)16-s − 0.552·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.13177 + 1.69024i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13177 + 1.69024i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-31.9 - 0.0155i)T \) |
good | 3 | \( 1 + (39.7 - 39.7i)T - 5.90e4iT^{2} \) |
| 5 | \( 1 + (4.09e3 - 4.09e3i)T - 9.76e6iT^{2} \) |
| 7 | \( 1 + 7.86e3T + 2.82e8T^{2} \) |
| 11 | \( 1 + (4.76e4 + 4.76e4i)T + 2.59e10iT^{2} \) |
| 13 | \( 1 + (-2.25e5 - 2.25e5i)T + 1.37e11iT^{2} \) |
| 17 | \( 1 + 7.84e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + (1.71e6 - 1.71e6i)T - 6.13e12iT^{2} \) |
| 23 | \( 1 - 1.21e7T + 4.14e13T^{2} \) |
| 29 | \( 1 + (-3.70e6 - 3.70e6i)T + 4.20e14iT^{2} \) |
| 31 | \( 1 + 3.97e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + (-6.77e6 + 6.77e6i)T - 4.80e15iT^{2} \) |
| 41 | \( 1 - 6.98e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + (-1.43e8 - 1.43e8i)T + 2.16e16iT^{2} \) |
| 47 | \( 1 - 2.76e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + (1.93e8 - 1.93e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 + (-4.10e8 - 4.10e8i)T + 5.11e17iT^{2} \) |
| 61 | \( 1 + (-2.74e8 - 2.74e8i)T + 7.13e17iT^{2} \) |
| 67 | \( 1 + (8.74e8 - 8.74e8i)T - 1.82e18iT^{2} \) |
| 71 | \( 1 - 2.68e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + 2.43e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 1.43e8iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (2.28e8 - 2.28e8i)T - 1.55e19iT^{2} \) |
| 89 | \( 1 - 1.16e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 4.04e9T + 7.37e19T^{2} \) |
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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
−16.49740065054637725159667461485, −15.57163803643705339335745480513, −14.51381513510995836745706549947, −13.07714312226702541882249545209, −11.34695158169468913182527694007, −10.77279787120341828104119947680, −7.74413018006127643865602848178, −6.42592648659972158268685849427, −4.25464838326942057491541221576, −2.83036510563786261583325087610,
0.71680837613946328080426567872, 3.55571871070489205995799203505, 4.98820747967921460178035512581, 6.93594056782735978444787717858, 8.700306275105210190148627088636, 11.15867033965053090058686197624, 12.45971875755720634588484195828, 13.02734906760109279973016749959, 15.23065473167252914910344474982, 15.82768680985611208565388607803