| L(s) = 1 | − 1.12·2-s − 2.92·3-s − 0.726·4-s + 2.92·5-s + 3.30·6-s + 3.07·8-s + 5.58·9-s − 3.30·10-s − 3.80·11-s + 2.12·12-s + 2.92·13-s − 8.58·15-s − 2.01·16-s − 2.23·17-s − 6.29·18-s − 6.13·19-s − 2.12·20-s + 4.28·22-s + 1.05·23-s − 9.01·24-s + 3.58·25-s − 3.30·26-s − 7.56·27-s + 3.63·29-s + 9.68·30-s + 6.54·31-s − 3.87·32-s + ⋯ |
| L(s) = 1 | − 0.797·2-s − 1.69·3-s − 0.363·4-s + 1.31·5-s + 1.34·6-s + 1.08·8-s + 1.86·9-s − 1.04·10-s − 1.14·11-s + 0.614·12-s + 0.812·13-s − 2.21·15-s − 0.504·16-s − 0.541·17-s − 1.48·18-s − 1.40·19-s − 0.475·20-s + 0.914·22-s + 0.220·23-s − 1.83·24-s + 0.716·25-s − 0.648·26-s − 1.45·27-s + 0.675·29-s + 1.76·30-s + 1.17·31-s − 0.685·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 3 | \( 1 + 2.92T + 3T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 6.13T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 - 3.63T + 29T^{2} \) |
| 31 | \( 1 - 6.54T + 31T^{2} \) |
| 37 | \( 1 + 9.35T + 37T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 + 1.77T + 47T^{2} \) |
| 53 | \( 1 + 0.934T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 9.53T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 1.86T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 7.70T + 97T^{2} \) |
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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
−8.730434032972671287722496622093, −8.253460924652166781656149792945, −6.88044874887887893987510196005, −6.45329409362611416793496973446, −5.48652888055762692082800915744, −5.06933145586121281558283843865, −4.13719106430963042701490959856, −2.27436390961543582730374358753, −1.20625702396805297936603425263, 0,
1.20625702396805297936603425263, 2.27436390961543582730374358753, 4.13719106430963042701490959856, 5.06933145586121281558283843865, 5.48652888055762692082800915744, 6.45329409362611416793496973446, 6.88044874887887893987510196005, 8.253460924652166781656149792945, 8.730434032972671287722496622093
