| L(s) = 1 | + (0.0433 − 1.41i)2-s + (0.739 + 1.78i)3-s + (−1.99 − 0.122i)4-s + (1.08 + 2.12i)5-s + (2.55 − 0.968i)6-s + (−0.938 + 3.90i)7-s + (−0.259 + 2.81i)8-s + (−0.521 + 0.521i)9-s + (3.05 − 1.43i)10-s + (−2.35 − 2.75i)11-s + (−1.25 − 3.65i)12-s + (2.54 − 4.15i)13-s + (5.48 + 1.49i)14-s + (−2.99 + 3.50i)15-s + (3.96 + 0.489i)16-s + (0.0520 − 0.661i)17-s + ⋯ |
| L(s) = 1 | + (0.0306 − 0.999i)2-s + (0.427 + 1.03i)3-s + (−0.998 − 0.0613i)4-s + (0.484 + 0.951i)5-s + (1.04 − 0.395i)6-s + (−0.354 + 1.47i)7-s + (−0.0918 + 0.995i)8-s + (−0.173 + 0.173i)9-s + (0.965 − 0.455i)10-s + (−0.709 − 0.830i)11-s + (−0.363 − 1.05i)12-s + (0.706 − 1.15i)13-s + (1.46 + 0.399i)14-s + (−0.773 + 0.905i)15-s + (0.992 + 0.122i)16-s + (0.0126 − 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20524 + 0.244008i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20524 + 0.244008i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0433 + 1.41i)T \) |
| 41 | \( 1 + (5.78 + 2.73i)T \) |
| good | 3 | \( 1 + (-0.739 - 1.78i)T + (-2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-1.08 - 2.12i)T + (-2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (0.938 - 3.90i)T + (-6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (2.35 + 2.75i)T + (-1.72 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.54 + 4.15i)T + (-5.90 - 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.0520 + 0.661i)T + (-16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-0.534 + 0.327i)T + (8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-3.84 + 2.79i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.0823 + 1.04i)T + (-28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (1.06 + 3.26i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.25 - 10.0i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (1.20 - 7.57i)T + (-40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (0.561 + 2.33i)T + (-41.8 + 21.3i)T^{2} \) |
| 53 | \( 1 + (-10.8 + 0.850i)T + (52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (3.02 + 4.15i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.97 - 12.4i)T + (-58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (9.59 + 8.19i)T + (10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-7.99 + 6.83i)T + (11.1 - 70.1i)T^{2} \) |
| 73 | \( 1 + (7.30 + 7.30i)T + 73iT^{2} \) |
| 79 | \( 1 + (-1.92 + 0.798i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 9.71iT - 83T^{2} \) |
| 89 | \( 1 + (4.79 + 1.15i)T + (79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (-1.98 - 1.69i)T + (15.1 + 95.8i)T^{2} \) |
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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
−12.92331850944518871361573932690, −11.72040119081140303365819503502, −10.62388789737048425705822944849, −10.14790674356325030039419188169, −9.061535328790396062778155425373, −8.351178844303846488102644102856, −6.10899338057460550526411096121, −5.09551091154358474082741286021, −3.25172327737745651203397466882, −2.77066156746790825761311082636,
1.36990364522275914110633743331, 4.02629310362175797672059564563, 5.25720587859508653613095393967, 6.87504996374937179598039576068, 7.26536933005779161564926488461, 8.447644271681466063213079160444, 9.398799488827960520994722635896, 10.51524390374011942093834824232, 12.41505066076950819022662441392, 13.21311817227413166490406145571
