L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.826 − 0.563i)5-s + (0.900 − 0.433i)8-s + (0.0747 + 0.997i)10-s + (0.988 − 0.149i)11-s + (−0.988 + 0.149i)13-s + (−0.900 − 0.433i)16-s + (0.733 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.733 − 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (0.733 + 0.680i)26-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.826 − 0.563i)5-s + (0.900 − 0.433i)8-s + (0.0747 + 0.997i)10-s + (0.988 − 0.149i)11-s + (−0.988 + 0.149i)13-s + (−0.900 − 0.433i)16-s + (0.733 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.733 − 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (0.733 + 0.680i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5444581692 - 0.7658959488i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5444581692 - 0.7658959488i\) |
\(L(1)\) |
\(\approx\) |
\(0.6210784230 - 0.2907256368i\) |
\(L(1)\) |
\(\approx\) |
\(0.6210784230 - 0.2907256368i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (-0.826 - 0.563i)T \) |
| 11 | \( 1 + (0.988 - 0.149i)T \) |
| 13 | \( 1 + (-0.988 + 0.149i)T \) |
| 17 | \( 1 + (0.733 - 0.680i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.955 + 0.294i)T \) |
| 29 | \( 1 + (0.733 - 0.680i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.955 + 0.294i)T \) |
| 41 | \( 1 + (-0.0747 + 0.997i)T \) |
| 43 | \( 1 + (0.0747 + 0.997i)T \) |
| 47 | \( 1 + (-0.623 - 0.781i)T \) |
| 53 | \( 1 + (-0.955 + 0.294i)T \) |
| 59 | \( 1 + (0.900 + 0.433i)T \) |
| 61 | \( 1 + (-0.222 - 0.974i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (-0.988 - 0.149i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.988 + 0.149i)T \) |
| 89 | \( 1 + (-0.365 - 0.930i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−24.060197149088397788501954548494, −23.52294496753151216392720777641, −22.4877407964652578042531777832, −21.90032039202745250018025339729, −20.28190062927844821027188794132, −19.41220816907418123074433433523, −19.14428291319311239885215213283, −17.88294023746279287084372214629, −17.233144580935789090955300233220, −16.29723866228655275770089921595, −15.41342685092159951717619541222, −14.64266962766849493490757098242, −14.109778527056400051458161423135, −12.54360663490643809591770479854, −11.63162858788019997706695637496, −10.57149954213229688569967207945, −9.79816506930515479851468322214, −8.66574957447646580767973679875, −7.83967643486089690962306217750, −6.949269334613775928573355807547, −6.22909142314614415880390265355, −4.85704089814403629774708804686, −3.894405345202230936312976906068, −2.36653791705368807952067095654, −0.81427740693462716499513521354,
0.45469543210596151611211241919, 1.54787598213422644805465202017, 2.93864611588514243231974319744, 4.00833022299116397234187380337, 4.782303505012599709680824275, 6.463688518938881018628187479393, 7.746568630107678887111939960720, 8.23410164150252183947707698147, 9.45134478138627816746509899292, 10.00954976831910639691361691039, 11.44408344150161474474577577336, 11.927551367288920925531969009849, 12.59516569802071476126259950570, 13.82060830591485513935444163368, 14.83319406197866308111220832679, 16.150144606032323830558859570426, 16.71045165067042423632669694712, 17.52315077306180787634334560037, 18.665811750003348855294056266881, 19.4535873765547419977827349474, 19.924713010660725072938615810909, 20.87090111585239096193460413278, 21.69162946627516288634377600348, 22.64498386892171074040761963735, 23.41809056031479302342619363416