L(s) = 1 | + 2-s + 0.414·3-s + 4-s + 5-s + 0.414·6-s − 7-s + 8-s − 2.82·9-s + 10-s + 0.414·12-s + 2.13·13-s − 14-s + 0.414·15-s + 16-s + 4.44·17-s − 2.82·18-s − 8.27·19-s + 20-s − 0.414·21-s − 0.646·23-s + 0.414·24-s + 25-s + 2.13·26-s − 2.41·27-s − 28-s + 0.428·29-s + 0.414·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.239·3-s + 0.5·4-s + 0.447·5-s + 0.169·6-s − 0.377·7-s + 0.353·8-s − 0.942·9-s + 0.316·10-s + 0.119·12-s + 0.591·13-s − 0.267·14-s + 0.106·15-s + 0.250·16-s + 1.07·17-s − 0.666·18-s − 1.89·19-s + 0.223·20-s − 0.0903·21-s − 0.134·23-s + 0.0845·24-s + 0.200·25-s + 0.418·26-s − 0.464·27-s − 0.188·28-s + 0.0796·29-s + 0.0756·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 13 | \( 1 - 2.13T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 8.27T + 19T^{2} \) |
| 23 | \( 1 + 0.646T + 23T^{2} \) |
| 29 | \( 1 - 0.428T + 29T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 + 8.49T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 - 0.0206T + 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 61 | \( 1 + 1.36T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 + 9.05T + 71T^{2} \) |
| 73 | \( 1 - 0.585T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 + 2.17T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.29943224893690894400538899764, −6.55870811316111722318936123646, −5.98953364147231876381699403842, −5.47579644128930334928571994644, −4.67466461003413361608446759902, −3.67994147396936548168109004166, −3.25550862994466498584217813827, −2.32549793410702565266513341996, −1.56624819212278216523012407055, 0,
1.56624819212278216523012407055, 2.32549793410702565266513341996, 3.25550862994466498584217813827, 3.67994147396936548168109004166, 4.67466461003413361608446759902, 5.47579644128930334928571994644, 5.98953364147231876381699403842, 6.55870811316111722318936123646, 7.29943224893690894400538899764