| L(s) = 1 | − i·2-s − 4-s + (−2 + i)5-s − 2i·7-s + i·8-s + (1 + 2i)10-s − 2i·13-s − 2·14-s + 16-s + 4·19-s + (2 − i)20-s + 2i·23-s + (3 − 4i)25-s − 2·26-s + 2i·28-s − 10·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.755i·7-s + 0.353i·8-s + (0.316 + 0.632i)10-s − 0.554i·13-s − 0.534·14-s + 0.250·16-s + 0.917·19-s + (0.447 − 0.223i)20-s + 0.417i·23-s + (0.600 − 0.800i)25-s − 0.392·26-s + 0.377i·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−8.267724809024335933901852634956, −7.45462785741029316642214847197, −7.14595469795646590654050690214, −5.84807965935127683751887173870, −5.03326406684402643340352897993, −3.93781936090595901129745507154, −3.57520941155622051122120816839, −2.58135226763692579673076694606, −1.22639578079610883399352430841, 0,
1.56068468832047221486785458455, 3.00412827139072834587607445520, 3.92401283534811723200182494521, 4.75422847809285521609863028506, 5.46569793647563722686364164625, 6.25013352262722179049166927489, 7.21812724275959140155326978769, 7.73747138244806281611040281017, 8.505560084999219944457161997324
