| L(s) = 1 | + 0.747·2-s + 3.07·3-s − 1.44·4-s − 5-s + 2.29·6-s + 3.23·7-s − 2.57·8-s + 6.45·9-s − 0.747·10-s − 2.73·11-s − 4.42·12-s + 4.31·13-s + 2.41·14-s − 3.07·15-s + 0.956·16-s + 4.82·18-s + 1.26·19-s + 1.44·20-s + 9.94·21-s − 2.04·22-s − 0.492·23-s − 7.91·24-s + 25-s + 3.22·26-s + 10.6·27-s − 4.65·28-s + 0.444·29-s + ⋯ |
| L(s) = 1 | + 0.528·2-s + 1.77·3-s − 0.720·4-s − 0.447·5-s + 0.938·6-s + 1.22·7-s − 0.909·8-s + 2.15·9-s − 0.236·10-s − 0.824·11-s − 1.27·12-s + 1.19·13-s + 0.646·14-s − 0.793·15-s + 0.239·16-s + 1.13·18-s + 0.291·19-s + 0.322·20-s + 2.16·21-s − 0.436·22-s − 0.102·23-s − 1.61·24-s + 0.200·25-s + 0.633·26-s + 2.04·27-s − 0.880·28-s + 0.0825·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.598247273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.598247273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.747T + 2T^{2} \) |
| 3 | \( 1 - 3.07T + 3T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 0.492T + 23T^{2} \) |
| 29 | \( 1 - 0.444T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 + 1.81T + 53T^{2} \) |
| 59 | \( 1 + 3.01T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 4.21T + 67T^{2} \) |
| 71 | \( 1 + 3.89T + 71T^{2} \) |
| 73 | \( 1 + 6.47T + 73T^{2} \) |
| 79 | \( 1 - 7.22T + 79T^{2} \) |
| 83 | \( 1 - 0.227T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−9.290824264043288182281981105873, −8.469609232826591955037385123653, −8.147836022068168120147067095545, −7.58064840198328418147950049294, −6.19100403381059681957792236832, −4.92594266740080162096959444058, −4.34734396431764754732375338439, −3.47849825985303737030577763010, −2.70148030880171466571022639414, −1.35847106090730897120407892690,
1.35847106090730897120407892690, 2.70148030880171466571022639414, 3.47849825985303737030577763010, 4.34734396431764754732375338439, 4.92594266740080162096959444058, 6.19100403381059681957792236832, 7.58064840198328418147950049294, 8.147836022068168120147067095545, 8.469609232826591955037385123653, 9.290824264043288182281981105873
