| L(s) = 1 | + 3-s − 2.64i·5-s + 3.74·7-s − 2·9-s + (−3 − 1.41i)11-s − 3.74·13-s − 2.64i·15-s − 4.24i·17-s − 4.24i·19-s + 3.74·21-s + 2.64i·23-s − 2.00·25-s − 5·27-s − 7.93i·31-s + (−3 − 1.41i)33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.18i·5-s + 1.41·7-s − 0.666·9-s + (−0.904 − 0.426i)11-s − 1.03·13-s − 0.683i·15-s − 1.02i·17-s − 0.973i·19-s + 0.816·21-s + 0.551i·23-s − 0.400·25-s − 0.962·27-s − 1.42i·31-s + (−0.522 − 0.246i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.651892707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.651892707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 2.64iT - 5T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 13 | \( 1 + 3.74T + 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 2.64iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 7.93iT - 31T^{2} \) |
| 37 | \( 1 - 7.93iT - 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 5.29iT - 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 + 2.64iT - 71T^{2} \) |
| 73 | \( 1 - 12.7iT - 73T^{2} \) |
| 79 | \( 1 - 7.48T + 79T^{2} \) |
| 83 | \( 1 + 8.48iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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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
−9.169346832973450289825768233804, −8.386529286002716938402962873035, −7.967946028538448167624553144459, −7.18669132775327247296090914788, −5.57907424085560933832084510535, −5.07509381311288574125915591253, −4.44363033886410528498633621582, −2.92954734687587399554473023486, −2.07525996161340417174861081891, −0.58165654258126435811752019268,
1.92721878585109086370964917757, 2.60833709984658275807877396922, 3.63966892832813977648407989884, 4.81279611166853636608127984457, 5.59846585159636654713331315460, 6.69517230560331973847997540862, 7.66348032432592018609554591567, 8.021503056057597589376503299801, 8.829528434523747449874745846816, 9.994224660028045406999204826234
