| L(s) = 1 | + (−1.34 + 1.08i)3-s − 3.01·5-s − 3.17·7-s + (0.630 − 2.93i)9-s − 4.05i·11-s − 2.81i·13-s + (4.05 − 3.27i)15-s − 2.19·17-s − 4.15·19-s + (4.27 − 3.45i)21-s + 4.69·23-s + 4.07·25-s + (2.34 + 4.63i)27-s + 2.09i·29-s − 5.12·31-s + ⋯ |
| L(s) = 1 | + (−0.777 + 0.628i)3-s − 1.34·5-s − 1.19·7-s + (0.210 − 0.977i)9-s − 1.22i·11-s − 0.779i·13-s + (1.04 − 0.846i)15-s − 0.532·17-s − 0.952·19-s + (0.933 − 0.753i)21-s + 0.977·23-s + 0.815·25-s + (0.450 + 0.892i)27-s + 0.389i·29-s − 0.921·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0150 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0150 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2308293192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2308293192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.34 - 1.08i)T \) |
| 167 | \( 1 + (-8.27 + 9.92i)T \) |
| good | 5 | \( 1 + 3.01T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 + 4.05iT - 11T^{2} \) |
| 13 | \( 1 + 2.81iT - 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 - 2.09iT - 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 7.67iT - 37T^{2} \) |
| 41 | \( 1 - 0.449T + 41T^{2} \) |
| 43 | \( 1 + 8.78iT - 43T^{2} \) |
| 47 | \( 1 - 8.78iT - 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.31T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 5.85T + 71T^{2} \) |
| 73 | \( 1 + 5.54iT - 73T^{2} \) |
| 79 | \( 1 + 2.67iT - 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 - 7.55iT - 89T^{2} \) |
| 97 | \( 1 + 0.448T + 97T^{2} \) |
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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
−9.097124387705271593928636163277, −8.874225858135343307159629147304, −7.71207543780186836546764479795, −6.92237701895247849946287083665, −6.12237369277465128746985133143, −5.42173085341618914198672348869, −4.29283754587402764926013927541, −3.62377496228766937847835844580, −2.97607773661252546574235070314, −0.61283137696121647777911079340,
0.16761902659562179861369476651, 1.80853053235656562744569141305, 3.05506600487384793639540409438, 4.24825878651613780346110963294, 4.69343766112425427018220203828, 6.00598409602749341890286902150, 6.87375169918552860130555007176, 7.08689801871842188531326524034, 8.005001997811713949684888620981, 8.917485670442303190409259460960
