L(s) = 1 | + (0.262 + 0.965i)2-s + (−0.862 + 0.505i)4-s + (−0.768 + 0.639i)5-s + (0.818 − 0.574i)7-s + (−0.714 − 0.699i)8-s + (−0.818 − 0.574i)10-s + (0.591 + 0.806i)11-s + (−0.0611 − 0.998i)13-s + (0.768 + 0.639i)14-s + (0.488 − 0.872i)16-s + (0.142 + 0.989i)17-s + (−0.862 + 0.505i)19-s + (0.339 − 0.940i)20-s + (−0.623 + 0.781i)22-s + (0.182 − 0.983i)25-s + (0.947 − 0.320i)26-s + ⋯ |
L(s) = 1 | + (0.262 + 0.965i)2-s + (−0.862 + 0.505i)4-s + (−0.768 + 0.639i)5-s + (0.818 − 0.574i)7-s + (−0.714 − 0.699i)8-s + (−0.818 − 0.574i)10-s + (0.591 + 0.806i)11-s + (−0.0611 − 0.998i)13-s + (0.768 + 0.639i)14-s + (0.488 − 0.872i)16-s + (0.142 + 0.989i)17-s + (−0.862 + 0.505i)19-s + (0.339 − 0.940i)20-s + (−0.623 + 0.781i)22-s + (0.182 − 0.983i)25-s + (0.947 − 0.320i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7798270661 + 1.294460066i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7798270661 + 1.294460066i\) |
\(L(1)\) |
\(\approx\) |
\(0.8763858601 + 0.6314349256i\) |
\(L(1)\) |
\(\approx\) |
\(0.8763858601 + 0.6314349256i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.262 + 0.965i)T \) |
| 5 | \( 1 + (-0.768 + 0.639i)T \) |
| 7 | \( 1 + (0.818 - 0.574i)T \) |
| 11 | \( 1 + (0.591 + 0.806i)T \) |
| 13 | \( 1 + (-0.0611 - 0.998i)T \) |
| 17 | \( 1 + (0.142 + 0.989i)T \) |
| 19 | \( 1 + (-0.862 + 0.505i)T \) |
| 31 | \( 1 + (0.992 - 0.122i)T \) |
| 37 | \( 1 + (0.794 - 0.607i)T \) |
| 41 | \( 1 + (0.415 - 0.909i)T \) |
| 43 | \( 1 + (-0.992 - 0.122i)T \) |
| 47 | \( 1 + (-0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.742 + 0.670i)T \) |
| 59 | \( 1 + (0.959 - 0.281i)T \) |
| 61 | \( 1 + (0.101 - 0.994i)T \) |
| 67 | \( 1 + (0.591 - 0.806i)T \) |
| 71 | \( 1 + (-0.882 + 0.470i)T \) |
| 73 | \( 1 + (0.301 + 0.953i)T \) |
| 79 | \( 1 + (0.488 + 0.872i)T \) |
| 83 | \( 1 + (-0.452 + 0.891i)T \) |
| 89 | \( 1 + (0.523 + 0.852i)T \) |
| 97 | \( 1 + (0.970 - 0.242i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.69130073301489123351645270113, −19.1355712988088634896897329300, −18.53388871362080513768950772502, −17.70436786680296303807125690681, −16.79917686349803590851781550731, −16.14596668542174307798643427432, −15.038900827244828654808850224851, −14.60723795906376066358123650944, −13.651522722401222064369500024914, −13.0864525275002179191299034379, −11.90663454196434199845232879668, −11.701733808886836311222648090464, −11.24700394165208232550031472578, −10.10768355891336921119559656305, −9.0929904940839451666944696296, −8.709706738375770681746016242567, −8.01465688499901902994846265104, −6.765662067432546027018886938814, −5.76122130947268711925394648761, −4.74711371800899373698433335418, −4.469166262708274745008670178397, −3.447661931031020459664477986434, −2.51474013189246453424630966581, −1.5485035042241323331243953043, −0.65540516073392324182158173659,
0.89674336711172021664135903980, 2.30426380762749743702995721171, 3.58805712654987518547631326517, 4.09677790847183452620705777306, 4.80891611119606439916423652190, 5.87631379114182858589898437634, 6.662522464206211170029484446415, 7.3961932611536494171904378449, 8.05582842415036865170807150525, 8.50049783096506238634339556376, 9.8155949235836103042880312726, 10.48874849371474430246160789812, 11.33088352806576169156457252498, 12.32311040607976844988375993352, 12.78798203316559894384946920396, 13.90671992766165439858069407342, 14.57141490186873377295648851837, 15.04111856256926421527374063466, 15.54524673541776234913109888301, 16.601296119339473032388609257228, 17.32169771141175766470741478303, 17.726167465733011873075360167775, 18.58833183474756844983160114323, 19.40068283858139235568790004915, 20.13460685887628911190242055296