L(s) = 1 | + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (0.222 + 0.974i)8-s + (0.222 − 0.974i)10-s + (0.900 + 0.433i)11-s + (0.900 + 0.433i)13-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 19-s + (0.623 − 0.781i)20-s + (0.623 + 0.781i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)26-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (0.222 + 0.974i)8-s + (0.222 − 0.974i)10-s + (0.900 + 0.433i)11-s + (0.900 + 0.433i)13-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 19-s + (0.623 − 0.781i)20-s + (0.623 + 0.781i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.772443044 + 0.4345505548i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.772443044 + 0.4345505548i\) |
\(L(1)\) |
\(\approx\) |
\(1.642380677 + 0.3096960682i\) |
\(L(1)\) |
\(\approx\) |
\(1.642380677 + 0.3096960682i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (-0.222 - 0.974i)T \) |
| 11 | \( 1 + (0.900 + 0.433i)T \) |
| 13 | \( 1 + (0.900 + 0.433i)T \) |
| 17 | \( 1 + (0.623 - 0.781i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.623 - 0.781i)T \) |
| 29 | \( 1 + (-0.623 + 0.781i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.623 - 0.781i)T \) |
| 41 | \( 1 + (-0.222 - 0.974i)T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (-0.900 - 0.433i)T \) |
| 53 | \( 1 + (-0.623 - 0.781i)T \) |
| 59 | \( 1 + (-0.222 + 0.974i)T \) |
| 61 | \( 1 + (-0.623 + 0.781i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.623 - 0.781i)T \) |
| 73 | \( 1 + (0.900 - 0.433i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.900 + 0.433i)T \) |
| 89 | \( 1 + (-0.900 + 0.433i)T \) |
| 97 | \( 1 - T \) |
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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
−28.08769773987223482091903194118, −27.4059189827604527703632855411, −25.930669565671033854271062518418, −25.15121149580987668791432551245, −23.78037364852270454225634367337, −23.15711832173001507602478853887, −22.093010013287606779620159339144, −21.4847030283637028266679016180, −20.194428238456768290122306873728, −19.247134951723463090958284345824, −18.47719395321929289606853121757, −16.943426250414852844220221705684, −15.586758102222255719451435555194, −14.78078556704341351600103515830, −13.90731568829166836511547367443, −12.80936707335502328418075876259, −11.54063712035055166657130778596, −10.87596990590084727652613097386, −9.74832654322376483101260603, −8.005778629481357377911186659119, −6.53610519664868786359024008674, −5.838092427764020517253532032460, −4.01377915069000616390472660979, −3.290242405745115157092872278020, −1.71043143061467810415429965102,
1.784375967298688477878931395309, 3.68540621357522467930698629452, 4.56116014635453763937885077734, 5.79043481381725513165128201904, 6.95992938503860655868101111500, 8.24733754589803007801150111357, 9.25420045444633604261414454870, 11.097886801537531757216029364434, 12.13962386051325338776155646445, 12.90043193064538043903077220860, 14.05273575022635708300334657786, 14.99487871866829806499958524279, 16.33467373101592103613286978441, 16.657440311006426032111806227, 18.05000659667752894439958875476, 19.64996325571030751507472957521, 20.546868651926471480221145300206, 21.31632983193704780744740876483, 22.51458291527984460104840026429, 23.41644880487070433700597476197, 24.22218631129441987243549559320, 25.1739899038728539659833151040, 25.88991582714351610506847542911, 27.37309350357051142178861534203, 28.23382595226720003964807060199