| L(s) = 1 | + 2.17·2-s + 3-s + 2.70·4-s − 2·5-s + 2.17·6-s + 1.53·8-s + 9-s − 4.34·10-s − 1.70·11-s + 2.70·12-s − 3.07·13-s − 2·15-s − 2.07·16-s + 2.04·17-s + 2.17·18-s + 6.34·19-s − 5.41·20-s − 3.70·22-s + 0.447·23-s + 1.53·24-s − 25-s − 6.68·26-s + 27-s + 2.63·29-s − 4.34·30-s − 5.75·31-s − 7.58·32-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.577·3-s + 1.35·4-s − 0.894·5-s + 0.885·6-s + 0.544·8-s + 0.333·9-s − 1.37·10-s − 0.515·11-s + 0.782·12-s − 0.853·13-s − 0.516·15-s − 0.519·16-s + 0.497·17-s + 0.511·18-s + 1.45·19-s − 1.21·20-s − 0.790·22-s + 0.0933·23-s + 0.314·24-s − 0.200·25-s − 1.31·26-s + 0.192·27-s + 0.488·29-s − 0.792·30-s − 1.03·31-s − 1.34·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 - 0.447T + 23T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 + 5.75T + 31T^{2} \) |
| 37 | \( 1 - 0.156T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 - 7.89T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 + 4.52T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 - 0.474T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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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
−7.20616562397674870834530409821, −6.83893262910560982179436224160, −5.69526782407752027226891494208, −5.18507945916305674008483448244, −4.58936427401265781779536654962, −3.78568858961027381419516806241, −3.23409432533015309645311555457, −2.70171346229910020998626370830, −1.61531866054342729101698203875, 0,
1.61531866054342729101698203875, 2.70171346229910020998626370830, 3.23409432533015309645311555457, 3.78568858961027381419516806241, 4.58936427401265781779536654962, 5.18507945916305674008483448244, 5.69526782407752027226891494208, 6.83893262910560982179436224160, 7.20616562397674870834530409821
