| L(s) = 1 | − 2i·5-s − 7-s + 2i·11-s + 2i·13-s − 2·17-s + 25-s − 8i·29-s + 8·31-s + 2i·35-s + 4i·37-s + 2·41-s + 10i·43-s − 4·47-s + 49-s − 12i·53-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s − 0.377·7-s + 0.603i·11-s + 0.554i·13-s − 0.485·17-s + 0.200·25-s − 1.48i·29-s + 1.43·31-s + 0.338i·35-s + 0.657i·37-s + 0.312·41-s + 1.52i·43-s − 0.583·47-s + 0.142·49-s − 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.645603419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.645603419\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.234478001984109873978363518907, −7.87033671697521534918003061363, −6.59106095295739766102385936769, −6.42490356413633172068184548476, −5.15458292471109153400824436071, −4.64052140343595092000810765717, −3.90958286248636377622440988773, −2.75313294668644967488786812973, −1.80592410161092673690940912608, −0.63122428606590345487515118807,
0.855076024767812872077051289248, 2.32574174193256105547932828870, 3.07558292377551450936890536266, 3.73445382354832769524019201901, 4.82143923734404707735838866992, 5.67308861390616284562900152792, 6.43352084217862655939081795922, 6.97396435540873469297528334742, 7.73556119101459360541781670367, 8.575686048537365608585292180295
