| L(s) = 1 | − 2.25·2-s − 0.0619·3-s + 3.07·4-s − 3.86·5-s + 0.139·6-s + 1.72·7-s − 2.41·8-s − 2.99·9-s + 8.70·10-s + 1.27·11-s − 0.190·12-s − 3.89·14-s + 0.239·15-s − 0.704·16-s + 7.17·17-s + 6.74·18-s + 1.11·19-s − 11.8·20-s − 0.107·21-s − 2.87·22-s + 9.40·23-s + 0.149·24-s + 9.95·25-s + 0.371·27-s + 5.31·28-s + 10.4·29-s − 0.539·30-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.0357·3-s + 1.53·4-s − 1.72·5-s + 0.0569·6-s + 0.653·7-s − 0.854·8-s − 0.998·9-s + 2.75·10-s + 0.384·11-s − 0.0549·12-s − 1.04·14-s + 0.0618·15-s − 0.176·16-s + 1.74·17-s + 1.59·18-s + 0.256·19-s − 2.65·20-s − 0.0233·21-s − 0.612·22-s + 1.96·23-s + 0.0305·24-s + 1.99·25-s + 0.0714·27-s + 1.00·28-s + 1.94·29-s − 0.0984·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6414607812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6414607812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 3 | \( 1 + 0.0619T + 3T^{2} \) |
| 5 | \( 1 + 3.86T + 5T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 17 | \( 1 - 7.17T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 - 9.40T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 37 | \( 1 - 3.69T + 37T^{2} \) |
| 41 | \( 1 + 7.35T + 41T^{2} \) |
| 43 | \( 1 + 7.03T + 43T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4.71T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 - 8.55T + 73T^{2} \) |
| 79 | \( 1 - 5.25T + 79T^{2} \) |
| 83 | \( 1 - 7.88T + 83T^{2} \) |
| 89 | \( 1 - 1.89T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.092750250499043051228436202053, −7.88071675348427387814841551831, −7.14196607442190561254073418071, −6.46874341582962330475354009355, −5.21360501721815452911982563071, −4.58732999204899383071591447815, −3.37101155479748263773604598663, −2.87386848526495853310579752611, −1.29589773860519632501271242671, −0.63932708765884878565687638680,
0.63932708765884878565687638680, 1.29589773860519632501271242671, 2.87386848526495853310579752611, 3.37101155479748263773604598663, 4.58732999204899383071591447815, 5.21360501721815452911982563071, 6.46874341582962330475354009355, 7.14196607442190561254073418071, 7.88071675348427387814841551831, 8.092750250499043051228436202053
