| L(s) = 1 | − 2.23·2-s − 1.89·3-s + 2.99·4-s − 2.70·5-s + 4.22·6-s − 2.21·8-s + 0.583·9-s + 6.04·10-s + 1.84·11-s − 5.66·12-s − 4.96·13-s + 5.11·15-s − 1.03·16-s + 4.51·17-s − 1.30·18-s − 0.717·19-s − 8.09·20-s − 4.12·22-s + 23-s + 4.19·24-s + 2.31·25-s + 11.0·26-s + 4.57·27-s + 9.20·29-s − 11.4·30-s − 4.49·31-s + 6.73·32-s + ⋯ |
| L(s) = 1 | − 1.57·2-s − 1.09·3-s + 1.49·4-s − 1.20·5-s + 1.72·6-s − 0.783·8-s + 0.194·9-s + 1.91·10-s + 0.557·11-s − 1.63·12-s − 1.37·13-s + 1.32·15-s − 0.257·16-s + 1.09·17-s − 0.307·18-s − 0.164·19-s − 1.80·20-s − 0.880·22-s + 0.208·23-s + 0.856·24-s + 0.462·25-s + 2.17·26-s + 0.880·27-s + 1.70·29-s − 2.08·30-s − 0.807·31-s + 1.19·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 1.89T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 + 4.96T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 + 0.717T + 19T^{2} \) |
| 29 | \( 1 - 9.20T + 29T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 - 6.26T + 37T^{2} \) |
| 41 | \( 1 - 4.59T + 41T^{2} \) |
| 43 | \( 1 - 0.827T + 43T^{2} \) |
| 47 | \( 1 + 5.92T + 47T^{2} \) |
| 53 | \( 1 + 7.86T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 3.90T + 79T^{2} \) |
| 83 | \( 1 + 0.345T + 83T^{2} \) |
| 89 | \( 1 + 1.04T + 89T^{2} \) |
| 97 | \( 1 + 6.55T + 97T^{2} \) |
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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
−9.533218662946602565938033575570, −8.512190863318396087024219803571, −7.83070120853453117004241374014, −7.16387162381011634568033430594, −6.39822623414664096624217250039, −5.19336858034297842009692483693, −4.24039324028195235273663151351, −2.76708548975325345282481466602, −1.04936795586857323648178376821, 0,
1.04936795586857323648178376821, 2.76708548975325345282481466602, 4.24039324028195235273663151351, 5.19336858034297842009692483693, 6.39822623414664096624217250039, 7.16387162381011634568033430594, 7.83070120853453117004241374014, 8.512190863318396087024219803571, 9.533218662946602565938033575570
