| L(s) = 1 | − 2.23·5-s − i·7-s + 2·11-s − 4.47i·13-s + 6.47i·17-s − 2·19-s − 4i·23-s + 5.00·25-s − 8.47·29-s − 0.472·31-s + 2.23i·35-s − 2.47i·37-s + 3.52·41-s − 2.47i·43-s + 6.47i·47-s + ⋯ |
| L(s) = 1 | − 0.999·5-s − 0.377i·7-s + 0.603·11-s − 1.24i·13-s + 1.56i·17-s − 0.458·19-s − 0.834i·23-s + 1.00·25-s − 1.57·29-s − 0.0847·31-s + 0.377i·35-s − 0.406i·37-s + 0.550·41-s − 0.376i·43-s + 0.944i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1824893564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1824893564\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.23T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 6.47iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 2.47iT - 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 2.47iT - 43T^{2} \) |
| 47 | \( 1 - 6.47iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 1.52iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.52iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4.94iT - 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + 3.52iT - 97T^{2} \) |
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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
−8.386522001947298041507748530244, −7.85090554725365587388513401452, −7.11270973800997754980526164641, −6.23023482819972037471261547844, −5.44393175134077289454331760122, −4.21733071223587649787686403617, −3.86584619562985714261504893541, −2.82512604558852880318999485056, −1.40149575098521119209871460294, −0.06317923134618627038059365220,
1.52329529041637748543082846219, 2.74685062295910314852673802033, 3.75491469732468528121836323025, 4.43424640636858183694546440247, 5.29541849731948431055297063946, 6.32543439080536584509533298283, 7.17517354104543404926330324015, 7.57110879679543954474598497254, 8.704942790286481871280063629772, 9.148506523573477664970490216631
