L(s) = 1 | + 2.07·2-s − 2.04·3-s + 2.31·4-s − 0.294·5-s − 4.23·6-s + 0.566·7-s + 0.649·8-s + 1.16·9-s − 0.611·10-s + 11-s − 4.71·12-s + 1.17·14-s + 0.600·15-s − 3.27·16-s + 1.40·17-s + 2.41·18-s − 5.25·19-s − 0.680·20-s − 1.15·21-s + 2.07·22-s − 3.30·23-s − 1.32·24-s − 4.91·25-s + 3.74·27-s + 1.31·28-s − 2.18·29-s + 1.24·30-s + ⋯ |
L(s) = 1 | + 1.46·2-s − 1.17·3-s + 1.15·4-s − 0.131·5-s − 1.72·6-s + 0.214·7-s + 0.229·8-s + 0.387·9-s − 0.193·10-s + 0.301·11-s − 1.36·12-s + 0.314·14-s + 0.155·15-s − 0.819·16-s + 0.341·17-s + 0.569·18-s − 1.20·19-s − 0.152·20-s − 0.252·21-s + 0.442·22-s − 0.689·23-s − 0.270·24-s − 0.982·25-s + 0.721·27-s + 0.247·28-s − 0.406·29-s + 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.07T + 2T^{2} \) |
| 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 + 0.294T + 5T^{2} \) |
| 7 | \( 1 - 0.566T + 7T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 + 2.18T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 + 2.65T + 37T^{2} \) |
| 41 | \( 1 + 4.07T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 + 4.59T + 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 2.78T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 4.29T + 73T^{2} \) |
| 79 | \( 1 - 7.13T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 9.74T + 97T^{2} \) |
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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
−8.781261926590600652582967368434, −7.85997355079436892333538992499, −6.70613696748225877578753693248, −6.20949556554553005881857308256, −5.57143960022198467465997082109, −4.75850741420515679082772054067, −4.14612011669385153124311902280, −3.14957836093502473829390655168, −1.85047959416570395408440925157, 0,
1.85047959416570395408440925157, 3.14957836093502473829390655168, 4.14612011669385153124311902280, 4.75850741420515679082772054067, 5.57143960022198467465997082109, 6.20949556554553005881857308256, 6.70613696748225877578753693248, 7.85997355079436892333538992499, 8.781261926590600652582967368434