| L(s) = 1 | − 0.732i·5-s + 7-s + 1.46i·11-s + 3.26i·13-s − 2·17-s − 4.73i·19-s − 3.46·23-s + 4.46·25-s − 5.46i·29-s + 4·31-s − 0.732i·35-s − 5.46i·37-s + 2·41-s − 1.46i·43-s + 10.9·47-s + ⋯ |
| L(s) = 1 | − 0.327i·5-s + 0.377·7-s + 0.441i·11-s + 0.906i·13-s − 0.485·17-s − 1.08i·19-s − 0.722·23-s + 0.892·25-s − 1.01i·29-s + 0.718·31-s − 0.123i·35-s − 0.898i·37-s + 0.312·41-s − 0.223i·43-s + 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.919550154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.919550154\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 + 0.732iT - 5T^{2} \) |
| 11 | \( 1 - 1.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.26iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4.73iT - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 5.46iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 5.46iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 1.46iT - 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 7.66iT - 59T^{2} \) |
| 61 | \( 1 - 13.1iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−8.543299953000330978440854876215, −7.61236088431634051358618390393, −7.01289901153516786372316720986, −6.24211999075239438322195710170, −5.38749594848974859794813310291, −4.43111780330977839070401810911, −4.17554785712838267347377461100, −2.71594448099081447169076654390, −2.00020415806930153266812922547, −0.75230275504825559925292742269,
0.851080905418928133302172024834, 2.04514786207183482324839957063, 3.04979352671262024151969069171, 3.76685215261090873670267718222, 4.78866649641898419718472458578, 5.50055941324693868980580841267, 6.29143854104491963904073160684, 6.95512906229478350307828635179, 7.979146626265892666153678690068, 8.241969483616414532705931974876
