| L(s) = 1 | + 2-s − 0.302·3-s + 4-s + 1.30·5-s − 0.302·6-s − 4.60·7-s + 8-s − 2.90·9-s + 1.30·10-s − 0.302·12-s + 2.30·13-s − 4.60·14-s − 0.394·15-s + 16-s + 6·17-s − 2.90·18-s − 2·19-s + 1.30·20-s + 1.39·21-s − 6.90·23-s − 0.302·24-s − 3.30·25-s + 2.30·26-s + 1.78·27-s − 4.60·28-s − 6.90·29-s − 0.394·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.174·3-s + 0.5·4-s + 0.582·5-s − 0.123·6-s − 1.74·7-s + 0.353·8-s − 0.969·9-s + 0.411·10-s − 0.0874·12-s + 0.638·13-s − 1.23·14-s − 0.101·15-s + 0.250·16-s + 1.45·17-s − 0.685·18-s − 0.458·19-s + 0.291·20-s + 0.304·21-s − 1.44·23-s − 0.0618·24-s − 0.660·25-s + 0.451·26-s + 0.344·27-s − 0.870·28-s − 1.28·29-s − 0.0720·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8954 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8954 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.193988621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.193988621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 41 | \( 1 - 0.908T + 41T^{2} \) |
| 43 | \( 1 - 6.60T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 8.69T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 5.21T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.71157866799394495605253979055, −6.66479809651979962489050042649, −6.22465368323738475170226821087, −5.74790021373330847389479407776, −5.30287619962338141236602974563, −3.88893333963176893407111324264, −3.62114167542665222126625851513, −2.76289523818575905946027685980, −2.02374908954706742347164089517, −0.61726113112337577104791094293,
0.61726113112337577104791094293, 2.02374908954706742347164089517, 2.76289523818575905946027685980, 3.62114167542665222126625851513, 3.88893333963176893407111324264, 5.30287619962338141236602974563, 5.74790021373330847389479407776, 6.22465368323738475170226821087, 6.66479809651979962489050042649, 7.71157866799394495605253979055
