L(s) = 1 | + (−0.901 − 0.655i)2-s + (−0.883 + 2.71i)3-s + (−0.234 − 0.720i)4-s + (−2.79 + 2.03i)5-s + (2.57 − 1.87i)6-s + (0.309 + 0.951i)7-s + (−0.949 + 2.92i)8-s + (−4.18 − 3.04i)9-s + 3.85·10-s + (3.31 − 0.0938i)11-s + 2.16·12-s + (1.66 + 1.21i)13-s + (0.344 − 1.05i)14-s + (−3.05 − 9.39i)15-s + (1.54 − 1.12i)16-s + (−1.56 + 1.13i)17-s + ⋯ |
L(s) = 1 | + (−0.637 − 0.463i)2-s + (−0.510 + 1.56i)3-s + (−0.117 − 0.360i)4-s + (−1.25 + 0.908i)5-s + (1.05 − 0.764i)6-s + (0.116 + 0.359i)7-s + (−0.335 + 1.03i)8-s + (−1.39 − 1.01i)9-s + 1.21·10-s + (0.999 − 0.0283i)11-s + 0.625·12-s + (0.462 + 0.335i)13-s + (0.0920 − 0.283i)14-s + (−0.788 − 2.42i)15-s + (0.386 − 0.280i)16-s + (−0.379 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.238054 + 0.346270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.238054 + 0.346270i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-3.31 + 0.0938i)T \) |
good | 2 | \( 1 + (0.901 + 0.655i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.883 - 2.71i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (2.79 - 2.03i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-1.66 - 1.21i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.56 - 1.13i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.501 + 1.54i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 0.807T + 23T^{2} \) |
| 29 | \( 1 + (-2.46 - 7.58i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.637 + 0.463i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.10 - 9.56i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.657 - 2.02i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 + (-2.33 + 7.19i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (8.75 + 6.36i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.01 + 3.13i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.871 + 0.632i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 2.40T + 67T^{2} \) |
| 71 | \( 1 + (-2.57 + 1.87i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.378 - 1.16i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.67 - 5.57i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-13.0 + 9.44i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 4.43T + 89T^{2} \) |
| 97 | \( 1 + (5.23 + 3.80i)T + (29.9 + 92.2i)T^{2} \) |
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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
−15.01956095650010782607057151528, −14.31678327579370015184437224040, −11.81199278298707664071522558902, −11.25621677964043527419582775240, −10.55461856290786464161793317974, −9.444589388585697563226796665217, −8.478992235353505226873407750052, −6.44454525568981877549202676158, −4.78739922937719734920564104709, −3.49036956941392011556591088240,
0.75956679774041450742393153840, 4.09718573179734231953423801607, 6.26109312490533133273424012004, 7.47236087200382735847346612896, 8.014593095135312942032151112706, 9.091096345077925969385782433464, 11.30092341230755457450175257677, 12.18721312245203477926076511245, 12.74653684214313439346246948129, 13.87386521536583530557639028546