L(s) = 1 | − 2-s + 1.73·3-s + 4-s − 5-s − 1.73·6-s − 7-s − 8-s + 10-s + 1.73·12-s + 5.73·13-s + 14-s − 1.73·15-s + 16-s + 0.732·17-s + 0.267·19-s − 20-s − 1.73·21-s − 5·23-s − 1.73·24-s + 25-s − 5.73·26-s − 5.19·27-s − 28-s − 8·29-s + 1.73·30-s − 2.73·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s − 0.707·6-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.500·12-s + 1.58·13-s + 0.267·14-s − 0.447·15-s + 0.250·16-s + 0.177·17-s + 0.0614·19-s − 0.223·20-s − 0.377·21-s − 1.04·23-s − 0.353·24-s + 0.200·25-s − 1.12·26-s − 1.00·27-s − 0.188·28-s − 1.48·29-s + 0.316·30-s − 0.490·31-s − 0.176·32-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 | \( 1 - 1.73T + 3T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 - 0.732T + 17T^{2} \) |
| 19 | \( 1 - 0.267T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 6.19T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 + 5.53T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 + 9.66T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 9.12T + 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 + 5T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.79808346476681715349104937063, −7.00116879226255352271909234808, −6.02455824115979350411303974925, −5.74643796539648033216495359455, −4.28446699869775921668057473024, −3.66330017643938442737818314132, −3.09196599858977200362851897482, −2.18913307032200890692767254056, −1.29438442649993723000930310360, 0,
1.29438442649993723000930310360, 2.18913307032200890692767254056, 3.09196599858977200362851897482, 3.66330017643938442737818314132, 4.28446699869775921668057473024, 5.74643796539648033216495359455, 6.02455824115979350411303974925, 7.00116879226255352271909234808, 7.79808346476681715349104937063