| L(s) = 1 | + i·3-s + 4i·7-s − 9-s − 11-s − 4i·13-s + 6i·17-s − 2·19-s − 4·21-s − i·27-s − 4·31-s − i·33-s + 10i·37-s + 4·39-s − 4i·43-s − 12i·47-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.51i·7-s − 0.333·9-s − 0.301·11-s − 1.10i·13-s + 1.45i·17-s − 0.458·19-s − 0.872·21-s − 0.192i·27-s − 0.718·31-s − 0.174i·33-s + 1.64i·37-s + 0.640·39-s − 0.609i·43-s − 1.75i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5836512780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5836512780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.944099753450140483701131653586, −8.406670940959611939016869668185, −7.88560461457428504278841830621, −6.64061738933164564092648977140, −5.79949382886829652868444939194, −5.44796420848256181208668250781, −4.51245541547634551866088709296, −3.46639760235553658048902488133, −2.73214859737152052898666180942, −1.73715924323968501689787815555,
0.17485245316416495548729879967, 1.31035520857349845371854116242, 2.38084222452896974483552278914, 3.47459118910935140510891112643, 4.34752464270713788699013988771, 5.01140666202599892525998246220, 6.19394791694086627396876091056, 6.81356815392152135847878409017, 7.53432765527172246837923659769, 7.83084687198399962992359028650
