| L(s) = 1 | − 2-s + 2.44·3-s + 4-s − 3.01·5-s − 2.44·6-s − 8-s + 2.96·9-s + 3.01·10-s + 2.12·11-s + 2.44·12-s − 1.81·13-s − 7.36·15-s + 16-s + 1.05·17-s − 2.96·18-s − 3.34·19-s − 3.01·20-s − 2.12·22-s + 6.27·23-s − 2.44·24-s + 4.09·25-s + 1.81·26-s − 0.0948·27-s − 3.63·29-s + 7.36·30-s + 0.813·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.40·3-s + 0.5·4-s − 1.34·5-s − 0.996·6-s − 0.353·8-s + 0.987·9-s + 0.953·10-s + 0.640·11-s + 0.704·12-s − 0.504·13-s − 1.90·15-s + 0.250·16-s + 0.255·17-s − 0.697·18-s − 0.768·19-s − 0.674·20-s − 0.452·22-s + 1.30·23-s − 0.498·24-s + 0.818·25-s + 0.356·26-s − 0.0182·27-s − 0.675·29-s + 1.34·30-s + 0.146·31-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
| \(L(1)\) |
\(\approx\) |
\(1.666246848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666246848\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 + 3.01T + 5T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 23 | \( 1 - 6.27T + 23T^{2} \) |
| 29 | \( 1 + 3.63T + 29T^{2} \) |
| 31 | \( 1 - 0.813T + 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 43 | \( 1 - 9.64T + 43T^{2} \) |
| 47 | \( 1 - 6.55T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 6.74T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 6.34T + 67T^{2} \) |
| 71 | \( 1 - 9.85T + 71T^{2} \) |
| 73 | \( 1 - 0.436T + 73T^{2} \) |
| 79 | \( 1 - 4.59T + 79T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.455589406689824259772479771831, −7.85711521848069854346408007573, −7.27976932130255914579361407183, −6.75038262474356447887161757754, −5.45807672656047200369681396516, −4.23894393985650837534461471135, −3.74313457535651595409557893427, −2.90411352145502976649127831712, −2.08862047376981057293731645325, −0.75960389197591574811017996442,
0.75960389197591574811017996442, 2.08862047376981057293731645325, 2.90411352145502976649127831712, 3.74313457535651595409557893427, 4.23894393985650837534461471135, 5.45807672656047200369681396516, 6.75038262474356447887161757754, 7.27976932130255914579361407183, 7.85711521848069854346408007573, 8.455589406689824259772479771831
