| L(s) = 1 | + 2-s − 1.76·3-s + 4-s + 1.49·5-s − 1.76·6-s + 3.65·7-s + 8-s + 0.127·9-s + 1.49·10-s − 3.80·11-s − 1.76·12-s + 1.15·13-s + 3.65·14-s − 2.63·15-s + 16-s − 6.90·17-s + 0.127·18-s − 1.36·19-s + 1.49·20-s − 6.46·21-s − 3.80·22-s + 23-s − 1.76·24-s − 2.77·25-s + 1.15·26-s + 5.08·27-s + 3.65·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.02·3-s + 0.5·4-s + 0.667·5-s − 0.721·6-s + 1.38·7-s + 0.353·8-s + 0.0423·9-s + 0.471·10-s − 1.14·11-s − 0.510·12-s + 0.319·13-s + 0.976·14-s − 0.681·15-s + 0.250·16-s − 1.67·17-s + 0.0299·18-s − 0.313·19-s + 0.333·20-s − 1.41·21-s − 0.810·22-s + 0.208·23-s − 0.360·24-s − 0.554·25-s + 0.226·26-s + 0.977·27-s + 0.690·28-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 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 + 6.90T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 + 4.95T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 6.12T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 + 9.55T + 71T^{2} \) |
| 73 | \( 1 - 0.982T + 73T^{2} \) |
| 79 | \( 1 - 5.33T + 79T^{2} \) |
| 83 | \( 1 - 3.48T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.65112798690928341567050138425, −6.64723698697619482572483983161, −6.22591911443288377933391306381, −5.44467660747759852443347664239, −4.75572375803013713451260780894, −4.65846659421466725692675965615, −3.20106161208439756304123061833, −2.22645580234863752672402295896, −1.54578541585865273620317791666, 0,
1.54578541585865273620317791666, 2.22645580234863752672402295896, 3.20106161208439756304123061833, 4.65846659421466725692675965615, 4.75572375803013713451260780894, 5.44467660747759852443347664239, 6.22591911443288377933391306381, 6.64723698697619482572483983161, 7.65112798690928341567050138425
