L(s) = 1 | + 2.82i·5-s + i·7-s − 1.41·11-s + 2·13-s + 7.07·23-s − 3.00·25-s + 9.89i·29-s − 4i·31-s − 2.82·35-s + 2.82i·41-s + 4i·43-s + 2.82·47-s − 49-s + 4.24i·53-s − 4.00i·55-s + ⋯ |
L(s) = 1 | + 1.26i·5-s + 0.377i·7-s − 0.426·11-s + 0.554·13-s + 1.47·23-s − 0.600·25-s + 1.83i·29-s − 0.718i·31-s − 0.478·35-s + 0.441i·41-s + 0.609i·43-s + 0.412·47-s − 0.142·49-s + 0.582i·53-s − 0.539i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.618727559\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.618727559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 8.48iT - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.733189071195541328252167845382, −7.897394527789769079431105920330, −7.06738684730086196521588923874, −6.67974968321078001454787863254, −5.75953146419699791890465920815, −5.09137582391419915812207194204, −3.99731226791336984636867128295, −3.05517010390007903827816977894, −2.61775185611172920938863115321, −1.29826683128971216750957846033,
0.50418021601403730046213203027, 1.39134891920474561152398552055, 2.58189510653713276519139774908, 3.69654397458359394624669619492, 4.45574951187842142800938981504, 5.17021160182888472689657312128, 5.79646367380113618093251967133, 6.79838716188303750090505015691, 7.49856804647380530578539035750, 8.452620728523920314772554795322