| L(s) = 1 | + (0.977 − 0.564i)2-s + (−1.96 + 3.40i)3-s + (−1.36 + 2.36i)4-s + (−1.81 − 3.14i)5-s + 4.43i·6-s + (0.995 + 6.92i)7-s + 7.59i·8-s + (−3.22 − 5.58i)9-s + (−3.54 − 2.04i)10-s + (1.00 + 10.9i)11-s + (−5.35 − 9.27i)12-s − 19.1i·13-s + (4.88 + 6.21i)14-s + 14.2·15-s + (−1.16 − 2.01i)16-s + (11.6 + 6.75i)17-s + ⋯ |
| L(s) = 1 | + (0.488 − 0.282i)2-s + (−0.655 + 1.13i)3-s + (−0.340 + 0.590i)4-s + (−0.362 − 0.628i)5-s + 0.739i·6-s + (0.142 + 0.989i)7-s + 0.949i·8-s + (−0.358 − 0.620i)9-s + (−0.354 − 0.204i)10-s + (0.0911 + 0.995i)11-s + (−0.446 − 0.772i)12-s − 1.47i·13-s + (0.348 + 0.443i)14-s + 0.950·15-s + (−0.0727 − 0.125i)16-s + (0.687 + 0.397i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.704204 + 0.835817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.704204 + 0.835817i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.995 - 6.92i)T \) |
| 11 | \( 1 + (-1.00 - 10.9i)T \) |
| good | 2 | \( 1 + (-0.977 + 0.564i)T + (2 - 3.46i)T^{2} \) |
| 3 | \( 1 + (1.96 - 3.40i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (1.81 + 3.14i)T + (-12.5 + 21.6i)T^{2} \) |
| 13 | \( 1 + 19.1iT - 169T^{2} \) |
| 17 | \( 1 + (-11.6 - 6.75i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-22.7 + 13.1i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-8.07 - 13.9i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 19.1iT - 841T^{2} \) |
| 31 | \( 1 + (13.1 - 22.7i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-27.1 - 46.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 23.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 38.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (1.89 + 3.28i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-36.0 + 62.4i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (27.1 - 47.0i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-52.8 + 30.5i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-28.5 + 49.4i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 34.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (98.9 + 57.1i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-69.8 + 40.3i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 56.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (12.7 + 22.1i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 63.8T + 9.40e3T^{2} \) |
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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
−14.78692308340678260566398791872, −13.10296191863382820704545304318, −12.26700772499575881182757050160, −11.52154871530017460073434118988, −10.15711073181620805270761462216, −9.012526736850530640495305421129, −7.81988993822877579998273785681, −5.33748989021901303256197646805, −4.85691633110918089877931537066, −3.29176399567749079666047183930,
0.963806960792574259312987824110, 3.93766110694356462292719685888, 5.69616911160870552926941823105, 6.75601252421293155046930077046, 7.55467078167943355531800011829, 9.523672104270187115484089959080, 10.95464357980020666564354300353, 11.74578109791262538671816813537, 13.08562414811034627184106934199, 13.99768256544895142927358046718
