| L(s) = 1 | + 2-s + 3.30·3-s + 4-s + 3.87·5-s + 3.30·6-s + 8-s + 7.94·9-s + 3.87·10-s − 6.05·11-s + 3.30·12-s − 2.22·13-s + 12.8·15-s + 16-s + 4.70·17-s + 7.94·18-s − 3.59·19-s + 3.87·20-s − 6.05·22-s + 0.533·23-s + 3.30·24-s + 9.99·25-s − 2.22·26-s + 16.3·27-s + 6.98·29-s + 12.8·30-s − 9.52·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s + 1.73·5-s + 1.35·6-s + 0.353·8-s + 2.64·9-s + 1.22·10-s − 1.82·11-s + 0.954·12-s − 0.617·13-s + 3.30·15-s + 0.250·16-s + 1.14·17-s + 1.87·18-s − 0.825·19-s + 0.865·20-s − 1.29·22-s + 0.111·23-s + 0.675·24-s + 1.99·25-s − 0.436·26-s + 3.14·27-s + 1.29·29-s + 2.33·30-s − 1.71·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.761307385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.761307385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 11 | \( 1 + 6.05T + 11T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 - 0.533T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 + 9.52T + 31T^{2} \) |
| 37 | \( 1 + 5.92T + 37T^{2} \) |
| 43 | \( 1 + 3.00T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 6.19T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 7.05T + 67T^{2} \) |
| 71 | \( 1 - 5.41T + 71T^{2} \) |
| 73 | \( 1 - 5.26T + 73T^{2} \) |
| 79 | \( 1 - 0.401T + 79T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 - 3.68T + 89T^{2} \) |
| 97 | \( 1 - 8.66T + 97T^{2} \) |
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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
−8.358286399776029056101893351024, −7.81402596971918792323522295314, −7.04814214029597671014140427918, −6.24848612794656726292053609087, −5.15728147995333065225718838311, −4.88307313809118932637426007676, −3.46931329183545669341853766231, −2.89090984441971631533693659201, −2.21121256554818720122840266651, −1.65703740192490960787302199229,
1.65703740192490960787302199229, 2.21121256554818720122840266651, 2.89090984441971631533693659201, 3.46931329183545669341853766231, 4.88307313809118932637426007676, 5.15728147995333065225718838311, 6.24848612794656726292053609087, 7.04814214029597671014140427918, 7.81402596971918792323522295314, 8.358286399776029056101893351024
