L(s) = 1 | + (−1.57 − 1.23i)2-s + (0.954 + 3.88i)4-s − 3.66·5-s − 1.93i·7-s + (3.29 − 7.29i)8-s + (5.76 + 4.51i)10-s + 0.752i·11-s − 13.0·13-s + (−2.38 + 3.04i)14-s + (−14.1 + 7.41i)16-s + 23.9·17-s + 4.35i·19-s + (−3.49 − 14.2i)20-s + (0.928 − 1.18i)22-s − 6.26i·23-s + ⋯ |
L(s) = 1 | + (−0.786 − 0.616i)2-s + (0.238 + 0.971i)4-s − 0.732·5-s − 0.276i·7-s + (0.411 − 0.911i)8-s + (0.576 + 0.451i)10-s + 0.0684i·11-s − 1.00·13-s + (−0.170 + 0.217i)14-s + (−0.886 + 0.463i)16-s + 1.40·17-s + 0.229i·19-s + (−0.174 − 0.711i)20-s + (0.0422 − 0.0538i)22-s − 0.272i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8082492921\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8082492921\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.57 + 1.23i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 5 | \( 1 + 3.66T + 25T^{2} \) |
| 7 | \( 1 + 1.93iT - 49T^{2} \) |
| 11 | \( 1 - 0.752iT - 121T^{2} \) |
| 13 | \( 1 + 13.0T + 169T^{2} \) |
| 17 | \( 1 - 23.9T + 289T^{2} \) |
| 23 | \( 1 + 6.26iT - 529T^{2} \) |
| 29 | \( 1 + 33.1T + 841T^{2} \) |
| 31 | \( 1 + 17.5iT - 961T^{2} \) |
| 37 | \( 1 - 41.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 5.51T + 1.68e3T^{2} \) |
| 43 | \( 1 - 84.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 18.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 41.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 69.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 87.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 105. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 74.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 48.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 95.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 65.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 3.38T + 7.92e3T^{2} \) |
| 97 | \( 1 + 23.5T + 9.40e3T^{2} \) |
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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
−10.08314108395384425372294029436, −9.726322151428736008780901968505, −8.593175350321748661681375708770, −7.56238045529165577365191507484, −7.43938609415844417313110959526, −5.90312240589764286756673405910, −4.46041295761632494531034174192, −3.58323736176239306030240292873, −2.43614286409568021185739503722, −0.895753647867246518157378979753,
0.51411816592709628896973315380, 2.16095329940405999303563058815, 3.66767020194969754007063190800, 5.05922588888340521775601744188, 5.77886669027525812573314941350, 7.05650806618666644249439179623, 7.62455567246623802501858287252, 8.376187198819928757004768598951, 9.385627428678342047824825526675, 10.00291221917951418279987664073