L(s) = 1 | − 2·4-s − 3.02·7-s − 3.39·13-s + 4·16-s + 7.92·19-s + 6.05·28-s + 0.572·31-s − 9.67·37-s + 13.0·43-s + 2.16·49-s + 6.79·52-s + 15.6·61-s − 8·64-s − 13.2·67-s + 16.2·73-s − 15.8·76-s − 6.94·79-s + 10.2·91-s + 7.15·97-s + 2.88·103-s − 14.1·109-s − 12.1·112-s + ⋯ |
L(s) = 1 | − 4-s − 1.14·7-s − 0.942·13-s + 16-s + 1.81·19-s + 1.14·28-s + 0.102·31-s − 1.59·37-s + 1.99·43-s + 0.308·49-s + 0.942·52-s + 1.99·61-s − 64-s − 1.61·67-s + 1.90·73-s − 1.81·76-s − 0.781·79-s + 1.07·91-s + 0.726·97-s + 0.284·103-s − 1.35·109-s − 1.14·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 + 3.02T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 0.572T + 31T^{2} \) |
| 37 | \( 1 + 9.67T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13.0T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.6T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 6.94T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7.15T + 97T^{2} \) |
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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
−7.66261891138762580363612084644, −7.22359540338385724284878038640, −6.31601913655788620083635213635, −5.44311732500634033150691218246, −5.01658066269666393028105144409, −3.97805215859693643000178196206, −3.35969187521112245702319085027, −2.55128146396278428022695371244, −1.06347145716813095971474512820, 0,
1.06347145716813095971474512820, 2.55128146396278428022695371244, 3.35969187521112245702319085027, 3.97805215859693643000178196206, 5.01658066269666393028105144409, 5.44311732500634033150691218246, 6.31601913655788620083635213635, 7.22359540338385724284878038640, 7.66261891138762580363612084644