| L(s) = 1 | + 2-s − 1.87·3-s + 4-s + 3.35·5-s − 1.87·6-s − 2.95·7-s + 8-s + 0.524·9-s + 3.35·10-s + 5.74·11-s − 1.87·12-s − 0.211·13-s − 2.95·14-s − 6.29·15-s + 16-s − 3.33·17-s + 0.524·18-s − 4.10·19-s + 3.35·20-s + 5.54·21-s + 5.74·22-s + 23-s − 1.87·24-s + 6.23·25-s − 0.211·26-s + 4.64·27-s − 2.95·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.08·3-s + 0.5·4-s + 1.49·5-s − 0.766·6-s − 1.11·7-s + 0.353·8-s + 0.174·9-s + 1.05·10-s + 1.73·11-s − 0.541·12-s − 0.0585·13-s − 0.789·14-s − 1.62·15-s + 0.250·16-s − 0.808·17-s + 0.123·18-s − 0.940·19-s + 0.749·20-s + 1.20·21-s + 1.22·22-s + 0.208·23-s − 0.383·24-s + 1.24·25-s − 0.0414·26-s + 0.894·27-s − 0.557·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 + 2.95T + 7T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 13 | \( 1 + 0.211T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + 4.10T + 19T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 + 8.21T + 31T^{2} \) |
| 37 | \( 1 + 9.64T + 37T^{2} \) |
| 41 | \( 1 + 4.87T + 41T^{2} \) |
| 43 | \( 1 + 3.88T + 43T^{2} \) |
| 47 | \( 1 - 0.850T + 47T^{2} \) |
| 53 | \( 1 - 2.79T + 53T^{2} \) |
| 59 | \( 1 - 7.22T + 59T^{2} \) |
| 61 | \( 1 - 3.30T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 + 1.54T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 + 9.54T + 79T^{2} \) |
| 83 | \( 1 - 3.09T + 83T^{2} \) |
| 89 | \( 1 + 0.672T + 89T^{2} \) |
| 97 | \( 1 - 1.90T + 97T^{2} \) |
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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
−7.08796349935650433608124660665, −6.66470168942628199086312953082, −6.27686003492395348057899447403, −5.63373576703230396729052369584, −5.14600689179740918384938623998, −4.05150081357815499884975114520, −3.41372637963363391508438202347, −2.20315526801503746597463371930, −1.52997470004133870697115714364, 0,
1.52997470004133870697115714364, 2.20315526801503746597463371930, 3.41372637963363391508438202347, 4.05150081357815499884975114520, 5.14600689179740918384938623998, 5.63373576703230396729052369584, 6.27686003492395348057899447403, 6.66470168942628199086312953082, 7.08796349935650433608124660665
