L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.415 + 0.909i)5-s + (0.841 − 0.540i)6-s + (−0.654 − 0.755i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.142 − 0.989i)10-s + (−0.142 − 0.989i)11-s + (−0.959 + 0.281i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)14-s + (−0.959 − 0.281i)15-s + (0.415 + 0.909i)16-s + (−0.142 − 0.989i)17-s + ⋯ |
L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.415 + 0.909i)5-s + (0.841 − 0.540i)6-s + (−0.654 − 0.755i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.142 − 0.989i)10-s + (−0.142 − 0.989i)11-s + (−0.959 + 0.281i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)14-s + (−0.959 − 0.281i)15-s + (0.415 + 0.909i)16-s + (−0.142 − 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3801367998 - 0.2217619259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3801367998 - 0.2217619259i\) |
\(L(1)\) |
\(\approx\) |
\(0.5208043153 + 0.02996662909i\) |
\(L(1)\) |
\(\approx\) |
\(0.5208043153 + 0.02996662909i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 11 | \( 1 + (-0.142 - 0.989i)T \) |
| 13 | \( 1 + (-0.959 - 0.281i)T \) |
| 17 | \( 1 + (-0.142 - 0.989i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.841 - 0.540i)T \) |
| 29 | \( 1 + (-0.959 + 0.281i)T \) |
| 31 | \( 1 + (-0.654 - 0.755i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.654 - 0.755i)T \) |
| 53 | \( 1 + (-0.959 + 0.281i)T \) |
| 59 | \( 1 + (0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (0.415 + 0.909i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.959 - 0.281i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.654 - 0.755i)T \) |
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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
−27.2877538454539413890137658664, −25.8582243358417360719004190916, −25.17477034405825304219574366383, −24.45426762207728085039319293095, −23.67908489620310632322131323473, −22.45124488679077891301640910068, −21.33117335901865769876489754841, −19.98769533950577101296534710315, −19.35339676676056386098742223161, −18.25981718086302455871147364732, −17.483424994487546585185037235802, −16.75395960058236397467691677329, −15.85881088988905317447652916088, −14.67333078014294164054132334713, −12.99587561713387173038736888008, −12.3818528339989743192909757698, −11.36800113589657521762695980960, −9.89190124105995873389043579338, −9.24051228083135986566322523957, −7.93999365098812147042486659091, −6.9638937444859738869032893634, −5.85613729544737841795968065828, −5.03563582838150768214965337201, −2.41779293428529460489673595662, −1.39622439644498700151079087260,
0.51590701472270335131932304444, 2.75189540806088129001763641829, 3.6121334466032882780461506937, 5.48395288806697400494034481185, 6.67306132059503732925575175066, 7.49160440477018639610979649891, 9.32076250619150415172842166489, 9.85286492573144567996879497610, 10.87118872825686355311944951481, 11.40351178158851712185189429028, 12.851220780793514920687019720167, 14.26074157391704590618966451968, 15.478520709591958628535393110411, 16.45329330141298618217610818553, 17.07389286770149024196178498448, 18.12236672619202594519064612144, 18.92906390252332392638133165908, 20.1181728770170257723100557229, 20.96884614691015143379072649885, 22.183418875306681202432175933305, 22.4798906848467128071822691744, 23.986190048489477762839585182202, 25.18712893304343426315030850889, 26.386767671210340328971674959, 26.7176529110260314185826769570