L(s) = 1 | + 6.78·7-s + 9.89i·11-s − 20.3·13-s + 19.1i·17-s − 12·19-s − 9.59i·23-s + 8.48i·29-s − 38·31-s + 6.78·37-s + 69.2i·41-s + 67.8·43-s − 76.7i·47-s − 3·49-s + 83.4i·59-s − 70·61-s + ⋯ |
L(s) = 1 | + 0.968·7-s + 0.899i·11-s − 1.56·13-s + 1.12i·17-s − 0.631·19-s − 0.417i·23-s + 0.292i·29-s − 1.22·31-s + 0.183·37-s + 1.69i·41-s + 1.57·43-s − 1.63i·47-s − 0.0612·49-s + 1.41i·59-s − 1.14·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.060973950\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060973950\) |
\(L(2)\) |
|
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 | \( 1 - 6.78T + 49T^{2} \) |
| 11 | \( 1 - 9.89iT - 121T^{2} \) |
| 13 | \( 1 + 20.3T + 169T^{2} \) |
| 17 | \( 1 - 19.1iT - 289T^{2} \) |
| 19 | \( 1 + 12T + 361T^{2} \) |
| 23 | \( 1 + 9.59iT - 529T^{2} \) |
| 29 | \( 1 - 8.48iT - 841T^{2} \) |
| 31 | \( 1 + 38T + 961T^{2} \) |
| 37 | \( 1 - 6.78T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 67.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 76.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 83.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70T + 3.72e3T^{2} \) |
| 67 | \( 1 + 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 13.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 30T + 6.24e3T^{2} \) |
| 83 | \( 1 - 134. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 32.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 94.9T + 9.40e3T^{2} \) |
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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
−10.26109228129343976091588761212, −9.404517058014126728409089157262, −8.463946259351285788846795000094, −7.64163296327016669517876548657, −6.97492945648656685065131804051, −5.77543101874315172434024204729, −4.79152775264693691450513299918, −4.15590735130681612774565201522, −2.54541287417446720637653757976, −1.61474841748215178196031196226,
0.32579053159555237919962059638, 1.93208644670924510571748527132, 3.01754997472860181795635429116, 4.38629594644032385999612074794, 5.14123772811048715190374210306, 6.03946445438410558535678368430, 7.37132274756840578362076882846, 7.71645344599473464635313574934, 8.905883993450082060130590817311, 9.467247361654180884766011177146