L(s) = 1 | − i·2-s − 4-s + (2 + i)5-s + i·8-s + (1 − 2i)10-s + 6·11-s − i·13-s + 16-s − 6·19-s + (−2 − i)20-s − 6i·22-s + 6i·23-s + (3 + 4i)25-s − 26-s + 2·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.894 + 0.447i)5-s + 0.353i·8-s + (0.316 − 0.632i)10-s + 1.80·11-s − 0.277i·13-s + 0.250·16-s − 1.37·19-s + (−0.447 − 0.223i)20-s − 1.27i·22-s + 1.25i·23-s + (0.600 + 0.800i)25-s − 0.196·26-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.982416784\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.982416784\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 - i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.866273678815799772379028715532, −9.050109140299954058908238931694, −8.424394013694957563115028387131, −7.03234729684659521237254278050, −6.37564738959050777857104004525, −5.49352698529555080795954766621, −4.29953957465948577963493537750, −3.44294927574075611608327444294, −2.26295244575655760729251193197, −1.26626032266537807946711357262,
1.08316327073093154542097857508, 2.40961131458569387104700177429, 4.11222484446782719110772750552, 4.58922386046172736197782588887, 6.00010299627369731866157783242, 6.29737015089268719595314095240, 7.16263961284405065907691354893, 8.439186645222602778280501127877, 8.935823235095889709165456676089, 9.565485540683676043501666521964