L(s) = 1 | + 2.36i·2-s + (2.69 − 1.32i)3-s − 1.59·4-s + 8.24i·5-s + (3.13 + 6.36i)6-s − 0.755·7-s + 5.68i·8-s + (5.49 − 7.12i)9-s − 19.5·10-s − 12.7i·11-s + (−4.29 + 2.11i)12-s − 1.31·13-s − 1.78i·14-s + (10.9 + 22.1i)15-s − 19.8·16-s + 16.7i·17-s + ⋯ |
L(s) = 1 | + 1.18i·2-s + (0.897 − 0.441i)3-s − 0.398·4-s + 1.64i·5-s + (0.521 + 1.06i)6-s − 0.107·7-s + 0.711i·8-s + (0.610 − 0.791i)9-s − 1.95·10-s − 1.15i·11-s + (−0.357 + 0.175i)12-s − 0.101·13-s − 0.127i·14-s + (0.727 + 1.47i)15-s − 1.23·16-s + 0.985i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.08343 + 1.73988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08343 + 1.73988i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.69 + 1.32i)T \) |
| 59 | \( 1 - 7.68iT \) |
good | 2 | \( 1 - 2.36iT - 4T^{2} \) |
| 5 | \( 1 - 8.24iT - 25T^{2} \) |
| 7 | \( 1 + 0.755T + 49T^{2} \) |
| 11 | \( 1 + 12.7iT - 121T^{2} \) |
| 13 | \( 1 + 1.31T + 169T^{2} \) |
| 17 | \( 1 - 16.7iT - 289T^{2} \) |
| 19 | \( 1 - 11.8T + 361T^{2} \) |
| 23 | \( 1 - 13.6iT - 529T^{2} \) |
| 29 | \( 1 + 36.0iT - 841T^{2} \) |
| 31 | \( 1 - 8.07T + 961T^{2} \) |
| 37 | \( 1 - 50.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 47.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 18.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 47.1iT - 2.80e3T^{2} \) |
| 61 | \( 1 - 41.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 84.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 100.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 60.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 72.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 171.T + 9.40e3T^{2} \) |
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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
−13.34148534889472962239506550418, −11.69473825338949309858365132401, −10.79182160106582847183804372941, −9.538676669384625918908789983685, −8.207755790389815712154439671992, −7.58561067188130564555066684056, −6.58223554549649274025763417019, −5.93094149889375455609794406157, −3.63074733395880393870251869147, −2.47426294597790973762715083960,
1.29734506867693379938292593084, 2.69053797308491730597044299742, 4.21467342501209265708460580392, 4.97711217299858251781558025550, 7.20375241295910271081060490976, 8.446623847548322808792962399667, 9.560323242777986214290788394756, 9.755735078240597325384056914155, 11.21246015876233337139298195992, 12.35879355754975548059366635332