L(s) = 1 | + 2-s + 1.24·3-s + 4-s − 5-s + 1.24·6-s − 2.49·7-s + 8-s − 1.44·9-s − 10-s − 5.80·11-s + 1.24·12-s − 2.49·14-s − 1.24·15-s + 16-s + 4.29·17-s − 1.44·18-s − 4.04·19-s − 20-s − 3.10·21-s − 5.80·22-s − 3.10·23-s + 1.24·24-s + 25-s − 5.54·27-s − 2.49·28-s − 5.60·29-s − 1.24·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.719·3-s + 0.5·4-s − 0.447·5-s + 0.509·6-s − 0.942·7-s + 0.353·8-s − 0.481·9-s − 0.316·10-s − 1.74·11-s + 0.359·12-s − 0.666·14-s − 0.321·15-s + 0.250·16-s + 1.04·17-s − 0.340·18-s − 0.928·19-s − 0.223·20-s − 0.678·21-s − 1.23·22-s − 0.648·23-s + 0.254·24-s + 0.200·25-s − 1.06·27-s − 0.471·28-s − 1.04·29-s − 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 1.24T + 3T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 + 8.67T + 73T^{2} \) |
| 79 | \( 1 + 1.82T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 0.454T + 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−8.800176385740764406766699162716, −7.985373795722477914738498992676, −7.56166198160634570414625617155, −6.39823167138376208597005228985, −5.66848780247796250289639394527, −4.77887482459473846261449749461, −3.59628703538594424196073394735, −3.05174755046420827900206684076, −2.20237866761952591688485711541, 0,
2.20237866761952591688485711541, 3.05174755046420827900206684076, 3.59628703538594424196073394735, 4.77887482459473846261449749461, 5.66848780247796250289639394527, 6.39823167138376208597005228985, 7.56166198160634570414625617155, 7.985373795722477914738498992676, 8.800176385740764406766699162716