| L(s) = 1 | − 0.545·2-s − 0.363·3-s − 1.70·4-s − 2.08·5-s + 0.198·6-s − 7-s + 2.01·8-s − 2.86·9-s + 1.13·10-s + 1.95·11-s + 0.618·12-s + 1.38·13-s + 0.545·14-s + 0.755·15-s + 2.30·16-s + 0.0704·17-s + 1.56·18-s − 7.23·19-s + 3.54·20-s + 0.363·21-s − 1.06·22-s − 8.04·23-s − 0.733·24-s − 0.672·25-s − 0.755·26-s + 2.13·27-s + 1.70·28-s + ⋯ |
| L(s) = 1 | − 0.385·2-s − 0.209·3-s − 0.851·4-s − 0.930·5-s + 0.0808·6-s − 0.377·7-s + 0.713·8-s − 0.955·9-s + 0.358·10-s + 0.588·11-s + 0.178·12-s + 0.384·13-s + 0.145·14-s + 0.195·15-s + 0.576·16-s + 0.0170·17-s + 0.368·18-s − 1.65·19-s + 0.792·20-s + 0.0792·21-s − 0.227·22-s − 1.67·23-s − 0.149·24-s − 0.134·25-s − 0.148·26-s + 0.410·27-s + 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2800124312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2800124312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 2 | \( 1 + 0.545T + 2T^{2} \) |
| 3 | \( 1 + 0.363T + 3T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 - 0.0704T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 8.04T + 23T^{2} \) |
| 29 | \( 1 + 6.77T + 29T^{2} \) |
| 31 | \( 1 + 0.278T + 31T^{2} \) |
| 37 | \( 1 - 2.26T + 37T^{2} \) |
| 41 | \( 1 - 2.08T + 41T^{2} \) |
| 43 | \( 1 - 7.28T + 43T^{2} \) |
| 47 | \( 1 + 0.0885T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 8.13T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 7.04T + 79T^{2} \) |
| 83 | \( 1 + 5.55T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 - 7.39T + 97T^{2} \) |
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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
−8.865045507396476036270484205782, −7.942213523885332016690555586660, −7.61923249499170527915068540826, −6.21155921238012405757430220870, −5.93533590592019544046006180509, −4.60677266494038960822778384050, −4.05505092671974265009763463964, −3.33898633290878622360032923622, −1.89988485112002520108556229565, −0.33645637159526278363009395645,
0.33645637159526278363009395645, 1.89988485112002520108556229565, 3.33898633290878622360032923622, 4.05505092671974265009763463964, 4.60677266494038960822778384050, 5.93533590592019544046006180509, 6.21155921238012405757430220870, 7.61923249499170527915068540826, 7.942213523885332016690555586660, 8.865045507396476036270484205782
