L(s) = 1 | + (−1 + 1.73i)3-s + (2 + 3.46i)5-s + (−0.499 − 0.866i)9-s − 7.99·15-s + (1 − 1.73i)17-s + (1 + 1.73i)19-s + (−4 − 6.92i)23-s + (−5.49 + 9.52i)25-s − 4.00·27-s + 2·29-s + (−2 + 3.46i)31-s + (3 + 5.19i)37-s − 2·41-s + 8·43-s + (1.99 − 3.46i)45-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.999i)3-s + (0.894 + 1.54i)5-s + (−0.166 − 0.288i)9-s − 2.06·15-s + (0.242 − 0.420i)17-s + (0.229 + 0.397i)19-s + (−0.834 − 1.44i)23-s + (−1.09 + 1.90i)25-s − 0.769·27-s + 0.371·29-s + (−0.359 + 0.622i)31-s + (0.493 + 0.854i)37-s − 0.312·41-s + 1.21·43-s + (0.298 − 0.516i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.551298 + 1.11210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.551298 + 1.11210i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.26038034934072338392547793775, −10.53419510080439711639113670991, −10.12280521541178391137915075361, −9.302809628078388426394940327673, −7.80608525003795736153474151825, −6.64816660922221670143116448464, −5.93037312169722568550874604180, −4.86117851223845245764222867140, −3.55683820040095531474443962281, −2.33708624739966318080004580333,
0.925685431336881532252264267021, 1.99503942564064382736952625075, 4.16126071596540159444221953544, 5.53675492852847305268946385047, 5.89617905358734567542833060655, 7.20560587115370710792019576716, 8.166102399136215303224807121059, 9.213776631061235487785156081965, 9.859162545584294594708466487086, 11.23285036828823323555144324239