| L(s) = 1 | − 2i·7-s + 6·11-s + 5i·13-s − 3i·17-s − 2·19-s − 6i·23-s + 3·29-s − 4·31-s − 5i·37-s + 6·41-s − 10i·43-s + 3·49-s + 6i·53-s − 12·59-s + 5·61-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s + 1.80·11-s + 1.38i·13-s − 0.727i·17-s − 0.458·19-s − 1.25i·23-s + 0.557·29-s − 0.718·31-s − 0.821i·37-s + 0.937·41-s − 1.52i·43-s + 0.428·49-s + 0.824i·53-s − 1.56·59-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.135532615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.135532615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 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
−7.50348794856445218778409891735, −6.89737847950395764792675495369, −6.55443922735829337686686927605, −5.75738444178253917026528819978, −4.55964410831323468102042976390, −4.23428773068078288403386023116, −3.59133829303834898944346915987, −2.40589687720989154030206189577, −1.55313423682174566059867730131, −0.57193006359597217657193650460,
1.03194512120367737414318965309, 1.82643417624337515951819697185, 2.93356796350294264714655670324, 3.59427819421124220355648968410, 4.35595291168250339697092642662, 5.25891364844804080467842390545, 6.00720067287797180623647173806, 6.35593895466780444428661170135, 7.28648595490524181190829612966, 8.021259038723229747414591210651
