L(s) = 1 | − 5-s + 0.807i·7-s + 2.90·11-s + 2.37·13-s + 6.47·17-s − 3.20i·19-s + (−2.82 + 3.87i)23-s + 25-s − 7.84i·29-s − 3.15·31-s − 0.807i·35-s + 7.68i·37-s − 0.186i·41-s + 2.61i·43-s − 10.5i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.305i·7-s + 0.876·11-s + 0.659·13-s + 1.57·17-s − 0.735i·19-s + (−0.589 + 0.807i)23-s + 0.200·25-s − 1.45i·29-s − 0.567·31-s − 0.136i·35-s + 1.26i·37-s − 0.0291i·41-s + 0.398i·43-s − 1.54i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.967564620\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.967564620\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (2.82 - 3.87i)T \) |
good | 7 | \( 1 - 0.807iT - 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 + 3.20iT - 19T^{2} \) |
| 29 | \( 1 + 7.84iT - 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 - 7.68iT - 37T^{2} \) |
| 41 | \( 1 + 0.186iT - 41T^{2} \) |
| 43 | \( 1 - 2.61iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 + 0.0710iT - 59T^{2} \) |
| 61 | \( 1 - 5.76iT - 61T^{2} \) |
| 67 | \( 1 - 2.04iT - 67T^{2} \) |
| 71 | \( 1 + 4.39iT - 71T^{2} \) |
| 73 | \( 1 + 4.19T + 73T^{2} \) |
| 79 | \( 1 + 0.308iT - 79T^{2} \) |
| 83 | \( 1 - 0.614T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 2.32iT - 97T^{2} \) |
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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
−8.367673204562301608782295033865, −7.74137880762136628362918373644, −7.01277819717497783029540190770, −6.13119676187242377585477024999, −5.57776968091952557477443274482, −4.57685291052852673603500697780, −3.74421179510125446287301435568, −3.13395575134763691951708877919, −1.87122468268561195978917853844, −0.812676065515661870268405096676,
0.860204814972473401826093697240, 1.78851368967311458460261875727, 3.23519218839981955830856819792, 3.73517746746698991490517040987, 4.52174466988116408725416531078, 5.57865124671073638830417344752, 6.16117028758167090873297131780, 7.06729543444193169214500466533, 7.67577033613492591987760335011, 8.379907059245791982788107434571