L(s) = 1 | + (−2.41 − 2.41i)5-s − 11.8·7-s + (11.9 − 11.9i)11-s + (−2.08 + 2.08i)13-s − 23.1·17-s + (6.77 + 6.77i)19-s + 3.92·23-s − 13.3i·25-s + (−0.782 + 0.782i)29-s + 2.65i·31-s + (28.6 + 28.6i)35-s + (−37.2 − 37.2i)37-s + 69.1i·41-s + (29.2 − 29.2i)43-s + 68.0i·47-s + ⋯ |
L(s) = 1 | + (−0.482 − 0.482i)5-s − 1.69·7-s + (1.08 − 1.08i)11-s + (−0.160 + 0.160i)13-s − 1.36·17-s + (0.356 + 0.356i)19-s + 0.170·23-s − 0.534i·25-s + (−0.0269 + 0.0269i)29-s + 0.0855i·31-s + (0.818 + 0.818i)35-s + (−1.00 − 1.00i)37-s + 1.68i·41-s + (0.681 − 0.681i)43-s + 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6357458168\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6357458168\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.41 + 2.41i)T + 25iT^{2} \) |
| 7 | \( 1 + 11.8T + 49T^{2} \) |
| 11 | \( 1 + (-11.9 + 11.9i)T - 121iT^{2} \) |
| 13 | \( 1 + (2.08 - 2.08i)T - 169iT^{2} \) |
| 17 | \( 1 + 23.1T + 289T^{2} \) |
| 19 | \( 1 + (-6.77 - 6.77i)T + 361iT^{2} \) |
| 23 | \( 1 - 3.92T + 529T^{2} \) |
| 29 | \( 1 + (0.782 - 0.782i)T - 841iT^{2} \) |
| 31 | \( 1 - 2.65iT - 961T^{2} \) |
| 37 | \( 1 + (37.2 + 37.2i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 69.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-29.2 + 29.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 68.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-40.3 - 40.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (23.5 - 23.5i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (65.9 - 65.9i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-40.9 - 40.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 98.1T + 5.04e3T^{2} \) |
| 73 | \( 1 - 74.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 0.779iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-91.7 - 91.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 20.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 86.6T + 9.40e3T^{2} \) |
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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
−9.519043631395203359817985759985, −9.074775980260177130102170830963, −8.344430950796512111315521013708, −7.12292305416512201107025971944, −6.44046861672151428150372145925, −5.76868256984624504244765447358, −4.34872685347957748257136556768, −3.67123083201005112965563087170, −2.68041512950077635909157168359, −0.917612944159444951648195089162,
0.24093562520436377834727628781, 2.08949619943235972883398739258, 3.29063115327360264498801193710, 3.94300722804981929087798789417, 5.08420557330609871917710368879, 6.51002233938296503468710180208, 6.74201712644669986915407532467, 7.51765701706610827535176246435, 8.915575947354355010387363553506, 9.371981669182734333987369037402