| L(s) = 1 | + (0.181 − 2.22i)5-s + 0.189i·7-s − 4.33·11-s + 4.67i·13-s + 0.0397i·17-s − 7.19·19-s − i·23-s + (−4.93 − 0.810i)25-s + 6.49·29-s + 9.17·31-s + (0.422 + 0.0344i)35-s + 3.03i·37-s + 8.69·41-s + 7.83i·43-s − 5.03i·47-s + ⋯ |
| L(s) = 1 | + (0.0812 − 0.996i)5-s + 0.0717i·7-s − 1.30·11-s + 1.29i·13-s + 0.00963i·17-s − 1.65·19-s − 0.208i·23-s + (−0.986 − 0.162i)25-s + 1.20·29-s + 1.64·31-s + (0.0714 + 0.00583i)35-s + 0.499i·37-s + 1.35·41-s + 1.19i·43-s − 0.733i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516302205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516302205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.181 + 2.22i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 0.189iT - 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 - 4.67iT - 13T^{2} \) |
| 17 | \( 1 - 0.0397iT - 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 - 9.17T + 31T^{2} \) |
| 37 | \( 1 - 3.03iT - 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 - 7.83iT - 43T^{2} \) |
| 47 | \( 1 + 5.03iT - 47T^{2} \) |
| 53 | \( 1 + 6.96iT - 53T^{2} \) |
| 59 | \( 1 - 4.64T + 59T^{2} \) |
| 61 | \( 1 - 7.86T + 61T^{2} \) |
| 67 | \( 1 - 2.52iT - 67T^{2} \) |
| 71 | \( 1 - 1.70T + 71T^{2} \) |
| 73 | \( 1 - 6.29iT - 73T^{2} \) |
| 79 | \( 1 - 9.82T + 79T^{2} \) |
| 83 | \( 1 + 0.417iT - 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 - 0.0584iT - 97T^{2} \) |
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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
−8.402599744277701642898152900800, −7.955614977594106788407148820332, −6.82392918293799448909875938006, −6.24753135518574570310879513084, −5.33862540833382905176863571475, −4.54755679395781094186968921271, −4.17308260476305041597921598654, −2.70455855389926844408689494548, −2.02080256007047414701446608683, −0.73525600562165748122563010504,
0.62812453968716277520107032226, 2.41122369872752836439622998954, 2.68878565436118723448773075720, 3.75835586262000026207851873300, 4.68016811311303851511195983271, 5.59375670133230270050246500352, 6.20983684483364871993296845785, 6.95655207052466084129118157583, 7.87188492031243551829833678856, 8.121963151371694784815566011266
