| L(s) = 1 | + 2-s − 2.80·3-s + 4-s + 1.81·5-s − 2.80·6-s + 1.41·7-s + 8-s + 4.85·9-s + 1.81·10-s − 3.53·11-s − 2.80·12-s − 5.56·13-s + 1.41·14-s − 5.09·15-s + 16-s − 7.46·17-s + 4.85·18-s + 5.03·19-s + 1.81·20-s − 3.96·21-s − 3.53·22-s + 1.78·23-s − 2.80·24-s − 1.69·25-s − 5.56·26-s − 5.18·27-s + 1.41·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.812·5-s − 1.14·6-s + 0.535·7-s + 0.353·8-s + 1.61·9-s + 0.574·10-s − 1.06·11-s − 0.808·12-s − 1.54·13-s + 0.378·14-s − 1.31·15-s + 0.250·16-s − 1.81·17-s + 1.14·18-s + 1.15·19-s + 0.406·20-s − 0.865·21-s − 0.754·22-s + 0.373·23-s − 0.571·24-s − 0.339·25-s − 1.09·26-s − 0.998·27-s + 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 - 1.81T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 3.53T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 + 0.387T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 - 6.27T + 71T^{2} \) |
| 73 | \( 1 + 0.960T + 73T^{2} \) |
| 79 | \( 1 - 2.78T + 79T^{2} \) |
| 83 | \( 1 + 9.07T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.246484092139936386226490056594, −7.76339858769947878503373524052, −7.10835284886719852785167740125, −6.30713096103530165223057610984, −5.46009599870760583267285018344, −5.04756588646568134113929928291, −4.43019465389198709604139727288, −2.72399753941570646001107465705, −1.74984707092054903729351227527, 0,
1.74984707092054903729351227527, 2.72399753941570646001107465705, 4.43019465389198709604139727288, 5.04756588646568134113929928291, 5.46009599870760583267285018344, 6.30713096103530165223057610984, 7.10835284886719852785167740125, 7.76339858769947878503373524052, 9.246484092139936386226490056594
