| L(s) = 1 | + i·3-s − i·5-s + 2.01·7-s − 9-s + 0.697·11-s + 0.195·13-s + 15-s − 0.430i·17-s − 7.71·19-s + 2.01i·21-s + (4.23 + 2.25i)23-s − 25-s − i·27-s − 9.64·29-s − 1.05i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s + 0.762·7-s − 0.333·9-s + 0.210·11-s + 0.0541·13-s + 0.258·15-s − 0.104i·17-s − 1.76·19-s + 0.440i·21-s + (0.882 + 0.471i)23-s − 0.200·25-s − 0.192i·27-s − 1.79·29-s − 0.188i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2796661019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2796661019\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 23 | \( 1 + (-4.23 - 2.25i)T \) |
| good | 7 | \( 1 - 2.01T + 7T^{2} \) |
| 11 | \( 1 - 0.697T + 11T^{2} \) |
| 13 | \( 1 - 0.195T + 13T^{2} \) |
| 17 | \( 1 + 0.430iT - 17T^{2} \) |
| 19 | \( 1 + 7.71T + 19T^{2} \) |
| 29 | \( 1 + 9.64T + 29T^{2} \) |
| 31 | \( 1 + 1.05iT - 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 + 10.6iT - 47T^{2} \) |
| 53 | \( 1 + 0.464iT - 53T^{2} \) |
| 59 | \( 1 - 7.64iT - 59T^{2} \) |
| 61 | \( 1 - 4.18iT - 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 + 5.69iT - 71T^{2} \) |
| 73 | \( 1 + 3.51T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 + 0.982T + 83T^{2} \) |
| 89 | \( 1 - 5.48iT - 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−7.993620196876160615724138468488, −7.20649175753298290211740273838, −6.38414167273624967042987835181, −5.47262227041723965637331041928, −4.99532636118370981445154282421, −4.14440987304338529032258116148, −3.59802413827644616840920616005, −2.33013006452974125135943022214, −1.53262189403246910163735270175, −0.06767359200602194762854321057,
1.43112115268009041176327969771, 2.12936864079785468307489115690, 3.08480374837815185771567638042, 4.03465117848130359079113504244, 4.83377954422486633810075271435, 5.63040267318921952538367835284, 6.58138379338753364065386190518, 6.81595749236602482682055859443, 7.891084032990956283342228740990, 8.221036030830318301077097696946
