| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·8-s − 9-s − 4·11-s + i·12-s + 2i·13-s + 16-s + 6i·17-s + i·18-s + 19-s + 4i·22-s − 4i·23-s + 24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s + 0.554i·13-s + 0.250·16-s + 1.45i·17-s + 0.235i·18-s + 0.229·19-s + 0.852i·22-s − 0.834i·23-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.424209793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424209793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 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
−8.552401287104038513288961730484, −8.055573476983164953450807104501, −7.26062068944332663136375686407, −6.27716716059072206147969104682, −5.60158468053906102238467638373, −4.61499983890732478160884711074, −3.79894750409145263244343700128, −2.66436393575797093320308151119, −2.03633990584666325228816963803, −0.76155585672793965265685803370,
0.69195047505261628641561516251, 2.57104854456918517039530813935, 3.29343566030573104969788660387, 4.47345439038734654798910190599, 5.13618679792012335541814618000, 5.67278318263461060108272592286, 6.65335616001666438648455720278, 7.59214014436594803960150618313, 7.965566884637014119472543713347, 8.920988396000801812360307286564
