| L(s) = 1 | + i·3-s + i·5-s − 0.279·7-s − 9-s − 0.628·11-s − 3.03·13-s − 15-s + 2.36i·17-s + 1.48·19-s − 0.279i·21-s + (1.08 − 4.67i)23-s − 25-s − i·27-s + 3.72·29-s + 8.99i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447i·5-s − 0.105·7-s − 0.333·9-s − 0.189·11-s − 0.840·13-s − 0.258·15-s + 0.573i·17-s + 0.341·19-s − 0.0609i·21-s + (0.226 − 0.974i)23-s − 0.200·25-s − 0.192i·27-s + 0.691·29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2986538969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2986538969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-1.08 + 4.67i)T \) |
| good | 7 | \( 1 + 0.279T + 7T^{2} \) |
| 11 | \( 1 + 0.628T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 - 2.36iT - 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 - 8.99iT - 31T^{2} \) |
| 37 | \( 1 + 5.99iT - 37T^{2} \) |
| 41 | \( 1 + 7.39T + 41T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 - 2.41iT - 47T^{2} \) |
| 53 | \( 1 - 1.22iT - 53T^{2} \) |
| 59 | \( 1 - 7.12iT - 59T^{2} \) |
| 61 | \( 1 + 3.34iT - 61T^{2} \) |
| 67 | \( 1 - 8.37T + 67T^{2} \) |
| 71 | \( 1 + 3.78iT - 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 + 12.8iT - 89T^{2} \) |
| 97 | \( 1 + 12.7iT - 97T^{2} \) |
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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
−8.133577239292181705999281289567, −7.04104337237237739153219760927, −6.68967285065048589920017065207, −5.69436799803709550855758217117, −4.98882773495036110357381340970, −4.33889915768456819442200868733, −3.32612609068095843639146105167, −2.77992498069902407053360530246, −1.67088766410399717856783944619, −0.080967666142078345130102281535,
1.09227620943827397466019202696, 2.11945771045850726173251889263, 2.96672673847083190646336003277, 3.86695684381710967937091910827, 5.05275416160520274754027967276, 5.22912001934640535786523764598, 6.38564547485867955012630817504, 6.90250435444592260588002436067, 7.76295508123822815501493258312, 8.147618934797611039167742220081
