L(s) = 1 | + (0.415 + 0.909i)2-s + (0.0713 − 0.997i)3-s + (−0.654 + 0.755i)4-s + (−0.281 + 0.959i)5-s + (0.936 − 0.349i)6-s + (−0.479 − 0.877i)7-s + (−0.959 − 0.281i)8-s + (−0.989 − 0.142i)9-s + (−0.989 + 0.142i)10-s + (0.959 − 0.281i)11-s + (0.707 + 0.707i)12-s + (−0.997 − 0.0713i)13-s + (0.599 − 0.800i)14-s + (0.936 + 0.349i)15-s + (−0.142 − 0.989i)16-s + (−0.909 − 0.415i)17-s + ⋯ |
L(s) = 1 | + (0.415 + 0.909i)2-s + (0.0713 − 0.997i)3-s + (−0.654 + 0.755i)4-s + (−0.281 + 0.959i)5-s + (0.936 − 0.349i)6-s + (−0.479 − 0.877i)7-s + (−0.959 − 0.281i)8-s + (−0.989 − 0.142i)9-s + (−0.989 + 0.142i)10-s + (0.959 − 0.281i)11-s + (0.707 + 0.707i)12-s + (−0.997 − 0.0713i)13-s + (0.599 − 0.800i)14-s + (0.936 + 0.349i)15-s + (−0.142 − 0.989i)16-s + (−0.909 − 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1831816354 - 0.3158812837i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1831816354 - 0.3158812837i\) |
\(L(1)\) |
\(\approx\) |
\(0.8001618971 + 0.1188056122i\) |
\(L(1)\) |
\(\approx\) |
\(0.8001618971 + 0.1188056122i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 3 | \( 1 + (0.0713 - 0.997i)T \) |
| 5 | \( 1 + (-0.281 + 0.959i)T \) |
| 7 | \( 1 + (-0.479 - 0.877i)T \) |
| 11 | \( 1 + (0.959 - 0.281i)T \) |
| 13 | \( 1 + (-0.997 - 0.0713i)T \) |
| 17 | \( 1 + (-0.909 - 0.415i)T \) |
| 19 | \( 1 + (-0.599 - 0.800i)T \) |
| 23 | \( 1 + (-0.800 + 0.599i)T \) |
| 29 | \( 1 + (0.877 - 0.479i)T \) |
| 31 | \( 1 + (-0.800 - 0.599i)T \) |
| 37 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (0.997 - 0.0713i)T \) |
| 43 | \( 1 + (0.877 + 0.479i)T \) |
| 47 | \( 1 + (0.755 + 0.654i)T \) |
| 53 | \( 1 + (-0.755 + 0.654i)T \) |
| 59 | \( 1 + (-0.0713 - 0.997i)T \) |
| 61 | \( 1 + (-0.977 - 0.212i)T \) |
| 67 | \( 1 + (-0.654 - 0.755i)T \) |
| 71 | \( 1 + (0.281 + 0.959i)T \) |
| 73 | \( 1 + (0.142 + 0.989i)T \) |
| 79 | \( 1 + (0.989 - 0.142i)T \) |
| 83 | \( 1 + (-0.936 + 0.349i)T \) |
| 97 | \( 1 + (-0.959 - 0.281i)T \) |
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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
−30.92994151308611233530709406007, −29.30171270608628129708573583524, −28.44353915777448556156067160288, −27.72379578697460771154637376304, −26.89222020358345709304386496440, −25.26815452305391665725736020685, −24.16072457558463858681259696397, −22.71226995979487200554488950592, −21.97626301265196023205294202273, −21.10877064182848956778843312694, −19.905860190995646277988341302970, −19.44581432065221471365651462369, −17.62235164908592676770958180638, −16.357793701121642863287594342076, −15.198304329601384605331451240425, −14.24117818534094400927309112668, −12.522419887947121852811408047037, −12.00010444658074406812149782278, −10.504037312991240945446367720045, −9.30144020675461988933056919582, −8.70513345071131424466580829915, −5.970420914006943853082927279412, −4.73824102048913784280744714949, −3.82055266817745228016017158351, −2.16601957202303590150489216281,
0.13774037018465449805985997959, 2.77710684574886797553608201831, 4.18644191899963260588469129486, 6.223324639082622136682631763105, 6.942498298214649707233203945687, 7.774449911382348834355916315794, 9.35791011569327400469128915968, 11.23601875293781571275337555609, 12.45989439857189983719118081693, 13.72522827215439487299937448500, 14.31424840865448932137067834973, 15.581994097488076343489867100250, 17.08919340661608031032923373581, 17.73208000470149080318964840325, 19.120200412840181890275450433448, 19.911520198258235771172234233651, 22.01376580819113991818097102524, 22.650102890096695615162790388193, 23.68051942158008112360408574798, 24.486739250586873046198439473331, 25.66436587611970566399706310084, 26.41952216909926221272485414642, 27.41962971608722371689437848886, 29.4420259430157164679725550670, 30.0086312300939820020443007733