| L(s) = 1 | − 3-s − 0.554i·5-s + 3.04i·7-s + 9-s − 1.80i·11-s + 0.554i·15-s − 1.24·17-s + 2.35i·19-s − 3.04i·21-s + 0.554·23-s + 4.69·25-s − 27-s − 6.18·29-s + 8.67i·31-s + 1.80i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.248i·5-s + 1.15i·7-s + 0.333·9-s − 0.543i·11-s + 0.143i·15-s − 0.302·17-s + 0.540i·19-s − 0.665i·21-s + 0.115·23-s + 0.938·25-s − 0.192·27-s − 1.14·29-s + 1.55i·31-s + 0.313i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7338607118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7338607118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.554iT - 5T^{2} \) |
| 7 | \( 1 - 3.04iT - 7T^{2} \) |
| 11 | \( 1 + 1.80iT - 11T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 - 2.35iT - 19T^{2} \) |
| 23 | \( 1 - 0.554T + 23T^{2} \) |
| 29 | \( 1 + 6.18T + 29T^{2} \) |
| 31 | \( 1 - 8.67iT - 31T^{2} \) |
| 37 | \( 1 + 0.960iT - 37T^{2} \) |
| 41 | \( 1 + 2.47iT - 41T^{2} \) |
| 43 | \( 1 + 0.384T + 43T^{2} \) |
| 47 | \( 1 + 9.96iT - 47T^{2} \) |
| 53 | \( 1 - 6.02T + 53T^{2} \) |
| 59 | \( 1 - 7.30iT - 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 13.6iT - 67T^{2} \) |
| 71 | \( 1 + 4.58iT - 71T^{2} \) |
| 73 | \( 1 + 1.04iT - 73T^{2} \) |
| 79 | \( 1 - 6.64T + 79T^{2} \) |
| 83 | \( 1 - 6.24iT - 83T^{2} \) |
| 89 | \( 1 - 13.4iT - 89T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
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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
−8.861999110404708172423910648990, −8.098352675452430146028734655814, −7.11502902758153823897067063377, −6.46266872598031804563159393546, −5.46067718714446399555736891649, −5.38318390796039887857912335807, −4.23538653520759228689607252505, −3.28347665855999370053463261072, −2.30623473283710752606156311575, −1.23042117288853037217171025010,
0.24492766536967416165419967273, 1.39622605626023436441090464220, 2.56847978261173307797403850151, 3.68647971440115129047331158309, 4.41639362709725636043674287074, 5.04172241608839079686544422855, 6.09245597844264469668653017579, 6.70048810180293937869923382455, 7.42854470532371579185417335231, 7.84235737140173559634529579022
