L(s) = 1 | − i·7-s + 3·11-s + i·13-s + 6i·17-s + 4·19-s + 3i·23-s − 3·29-s + 5·31-s + 2i·37-s + 3·41-s + i·43-s − 9i·47-s + 6·49-s + 6i·53-s + 3·59-s + ⋯ |
L(s) = 1 | − 0.377i·7-s + 0.904·11-s + 0.277i·13-s + 1.45i·17-s + 0.917·19-s + 0.625i·23-s − 0.557·29-s + 0.898·31-s + 0.328i·37-s + 0.468·41-s + 0.152i·43-s − 1.31i·47-s + 0.857·49-s + 0.824i·53-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.994907320\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.994907320\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 11iT - 97T^{2} \) |
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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
−7.84338112596323150882631883631, −7.31039278762939783889507413057, −6.52274475952408488530676944539, −5.96730261744021424380973256135, −5.19860307427021678788168853158, −4.20129086289801032209841002749, −3.81160464464791445887156705222, −2.90669552964214996392746125842, −1.75054148625559444586817797080, −1.05931053526598200436593927352,
0.52672668445655930250360739169, 1.53180285758089839163319764635, 2.67195009131145254640968682357, 3.19955608465450314044562727873, 4.26686782263029816765606483409, 4.83281705399450152420994635044, 5.72575710616252760993401555190, 6.21663477271828957542623695498, 7.22247895324900499143992991246, 7.46301477222446086731066848235