L(s) = 1 | + 3.41·7-s + 2.58·11-s + 3.41i·13-s + 1.17·17-s − 4.82i·23-s + 6i·29-s − 6.48i·31-s − 9.07i·37-s − 11.0i·41-s + 6.82·43-s − 5.65i·47-s + 4.65·49-s − 1.17·53-s − 6.58·59-s + 12.8·61-s + ⋯ |
L(s) = 1 | + 1.29·7-s + 0.779·11-s + 0.946i·13-s + 0.284·17-s − 1.00i·23-s + 1.11i·29-s − 1.16i·31-s − 1.49i·37-s − 1.72i·41-s + 1.04·43-s − 0.825i·47-s + 0.665·49-s − 0.160·53-s − 0.857·59-s + 1.64·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.668758379\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.668758379\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 - 3.41iT - 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 6.48iT - 31T^{2} \) |
| 37 | \( 1 + 9.07iT - 37T^{2} \) |
| 41 | \( 1 + 11.0iT - 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 + 5.65iT - 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.17iT - 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 - 2.48iT - 97T^{2} \) |
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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
−7.894894016790121202177878452985, −7.16576728647461623712356047543, −6.64031957934712246084302908193, −5.66887411230918146196161380073, −5.09980137353921805391568275552, −4.15787296564755641519441293124, −3.85225060863565096372037969284, −2.41689080389058408185459328496, −1.82030536865081650743799098372, −0.78285196290104143067863298077,
1.00213023349680534188596210569, 1.65468055549536839268175530670, 2.79300740564406019283127412599, 3.58978304925506452885893240226, 4.51852655636258617213760455770, 5.04930551926377605564077351087, 5.82796177566543245858192883201, 6.52513515650768662092216050147, 7.41404782442675803316194310703, 8.080244582989637649329689377656