L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s − 3.41i·11-s + (−1.43 + 1.43i)13-s + 1.00·14-s − 1.00·16-s + (−1.54 + 1.54i)17-s + 2.04i·19-s + (2.41 + 2.41i)22-s + (−1.15 − 1.15i)23-s − 2.03i·26-s + (−0.707 + 0.707i)28-s + 8.04·29-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s − 1.02i·11-s + (−0.399 + 0.399i)13-s + 0.267·14-s − 0.250·16-s + (−0.374 + 0.374i)17-s + 0.470i·19-s + (0.514 + 0.514i)22-s + (−0.241 − 0.241i)23-s − 0.399i·26-s + (−0.133 + 0.133i)28-s + 1.49·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4283694933\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4283694933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (1.43 - 1.43i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.54 - 1.54i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.04iT - 19T^{2} \) |
| 23 | \( 1 + (1.15 + 1.15i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.04T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 + (-0.0498 - 0.0498i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.94iT - 41T^{2} \) |
| 43 | \( 1 + (4.65 - 4.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.39 - 1.39i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.97 - 2.97i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.25T + 59T^{2} \) |
| 61 | \( 1 + 6.88T + 61T^{2} \) |
| 67 | \( 1 + (-5.84 - 5.84i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.92iT - 71T^{2} \) |
| 73 | \( 1 + (5.97 - 5.97i)T - 73iT^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 + (1.52 + 1.52i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.72T + 89T^{2} \) |
| 97 | \( 1 + (-9.64 - 9.64i)T + 97iT^{2} \) |
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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
−8.836151615536080799999595552005, −8.332324978709654261122581904572, −7.52820787837167349354395443708, −6.74242145322393186929126203577, −6.14055254111697165725536860250, −5.36500824980677515975457444842, −4.39117815858650477910046557788, −3.50133597726714942950528179179, −2.39115011807087687853522091842, −1.14695769147468203588722410846,
0.16804208534377308460233131434, 1.65112668303865168361135161776, 2.55889692384374944975217946092, 3.36796787337537526189161239719, 4.50514821194626199178886094230, 5.09968287326740607800764297732, 6.26291142461834052789676441593, 7.02211461832187014366466963727, 7.66396667083030004675983560103, 8.486648666043558866857034624056