L(s) = 1 | − 2-s + 4-s − 3.02i·5-s − 7-s − 8-s + 3.02i·10-s + 4.70i·11-s − 6.26i·13-s + 14-s + 16-s + 4.70i·17-s + (4.28 − 0.781i)19-s − 3.02i·20-s − 4.70i·22-s − 0.460i·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.35i·5-s − 0.377·7-s − 0.353·8-s + 0.958i·10-s + 1.41i·11-s − 1.73i·13-s + 0.267·14-s + 0.250·16-s + 1.14i·17-s + (0.983 − 0.179i)19-s − 0.677i·20-s − 1.00i·22-s − 0.0961i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9810252018\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9810252018\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + (-4.28 + 0.781i)T \) |
good | 5 | \( 1 + 3.02iT - 5T^{2} \) |
| 11 | \( 1 - 4.70iT - 11T^{2} \) |
| 13 | \( 1 + 6.26iT - 13T^{2} \) |
| 17 | \( 1 - 4.70iT - 17T^{2} \) |
| 23 | \( 1 + 0.460iT - 23T^{2} \) |
| 29 | \( 1 - 3.18T + 29T^{2} \) |
| 31 | \( 1 + 7.73iT - 31T^{2} \) |
| 37 | \( 1 - 1.81iT - 37T^{2} \) |
| 41 | \( 1 - 8.40T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + 12.5iT - 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 12.0iT - 67T^{2} \) |
| 71 | \( 1 + 2.20T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 8.69iT - 79T^{2} \) |
| 83 | \( 1 + 8.37iT - 83T^{2} \) |
| 89 | \( 1 + 6.19T + 89T^{2} \) |
| 97 | \( 1 + 9.50iT - 97T^{2} \) |
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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
−8.793792113330380316254216983790, −7.85242544851765325202966410155, −7.64092260100435267429365405414, −6.39849849320740745395317731481, −5.57472061690823923402085056269, −4.84756206189057821908209802549, −3.87207083798501121573397564669, −2.67006007725121799259722691201, −1.50573814534039013815852795211, −0.47281953770340901821317528367,
1.20426644388036515099877312123, 2.71579979901255716315723281247, 3.09180152778318253522117455524, 4.23350083189142087138304421298, 5.59170857253789699225958934072, 6.38502133170922746103016013028, 6.95781824440853177973732823900, 7.52576850309698475698182067293, 8.554653004185462276333526197413, 9.315220268180533370408669805188