L(s) = 1 | + (−0.947 + 0.319i)2-s + (−0.856 + 0.515i)3-s + (0.796 − 0.605i)4-s + (−0.370 − 0.928i)5-s + (0.647 − 0.762i)6-s + (0.907 + 0.419i)7-s + (−0.561 + 0.827i)8-s + (0.468 − 0.883i)9-s + (0.647 + 0.762i)10-s + (0.267 + 0.963i)11-s + (−0.370 + 0.928i)12-s + (0.468 + 0.883i)13-s + (−0.994 − 0.108i)14-s + (0.796 + 0.605i)15-s + (0.267 − 0.963i)16-s + (0.907 − 0.419i)17-s + ⋯ |
L(s) = 1 | + (−0.947 + 0.319i)2-s + (−0.856 + 0.515i)3-s + (0.796 − 0.605i)4-s + (−0.370 − 0.928i)5-s + (0.647 − 0.762i)6-s + (0.907 + 0.419i)7-s + (−0.561 + 0.827i)8-s + (0.468 − 0.883i)9-s + (0.647 + 0.762i)10-s + (0.267 + 0.963i)11-s + (−0.370 + 0.928i)12-s + (0.468 + 0.883i)13-s + (−0.994 − 0.108i)14-s + (0.796 + 0.605i)15-s + (0.267 − 0.963i)16-s + (0.907 − 0.419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4810410967 + 0.1437985134i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4810410967 + 0.1437985134i\) |
\(L(1)\) |
\(\approx\) |
\(0.5827605755 + 0.1203877576i\) |
\(L(1)\) |
\(\approx\) |
\(0.5827605755 + 0.1203877576i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (-0.947 + 0.319i)T \) |
| 3 | \( 1 + (-0.856 + 0.515i)T \) |
| 5 | \( 1 + (-0.370 - 0.928i)T \) |
| 7 | \( 1 + (0.907 + 0.419i)T \) |
| 11 | \( 1 + (0.267 + 0.963i)T \) |
| 13 | \( 1 + (0.468 + 0.883i)T \) |
| 17 | \( 1 + (0.907 - 0.419i)T \) |
| 19 | \( 1 + (0.976 + 0.214i)T \) |
| 23 | \( 1 + (-0.161 - 0.986i)T \) |
| 29 | \( 1 + (-0.947 - 0.319i)T \) |
| 31 | \( 1 + (0.976 - 0.214i)T \) |
| 37 | \( 1 + (-0.561 - 0.827i)T \) |
| 41 | \( 1 + (-0.161 + 0.986i)T \) |
| 43 | \( 1 + (0.267 - 0.963i)T \) |
| 47 | \( 1 + (-0.370 + 0.928i)T \) |
| 53 | \( 1 + (0.647 - 0.762i)T \) |
| 61 | \( 1 + (-0.947 + 0.319i)T \) |
| 67 | \( 1 + (-0.561 + 0.827i)T \) |
| 71 | \( 1 + (-0.370 + 0.928i)T \) |
| 73 | \( 1 + (-0.994 - 0.108i)T \) |
| 79 | \( 1 + (-0.856 - 0.515i)T \) |
| 83 | \( 1 + (0.0541 - 0.998i)T \) |
| 89 | \( 1 + (-0.947 - 0.319i)T \) |
| 97 | \( 1 + (-0.994 + 0.108i)T \) |
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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
−33.1233855198245605461651058574, −30.88517603571541791642382804245, −30.02367198980436425680083848274, −29.509983915402995832575125410895, −27.89383780028390029454425229426, −27.35274318367125740918032979556, −26.182700041301945658781725975287, −24.75435074054093054308131333774, −23.699422088225342786813394335680, −22.38442894462675663319788435579, −21.21053509881362190535229742566, −19.67620933782080119264969140291, −18.608438490174901194688327588180, −17.844405247318960849001058129742, −16.79594034678165702421816513652, −15.4698396170330154686841361570, −13.71619846317767177746114549423, −11.922636246405650836151457553753, −11.1426935838773058168107889517, −10.28819653909058379421003197498, −8.11945821743760865997701134078, −7.29074080507780680541349654216, −5.81975128215036787599464041424, −3.361155411255152986445977162409, −1.2649619380716004988907591948,
1.41508389210653202404743129866, 4.52663778727195484491999916035, 5.709996476243771918569419235612, 7.403559896790828203259967344090, 8.84180381234602113411721699471, 9.90100807435285747758255230035, 11.48898282537664414880402342430, 12.10844623148261610148130917534, 14.605434850187744224338651598890, 15.81687435378924429864381827508, 16.66851460382872340658882986781, 17.66391160129951873074899668929, 18.727065103207629702178093838189, 20.47802384393614863684915040924, 21.03505221739754668382245923813, 22.908566151897682290518204204130, 24.0235493409905876074953790233, 24.85152126969168887461689485950, 26.41111006356290135076425767207, 27.5538596472241345355851937328, 28.14800070144266530333086320620, 28.850472088992155354519468217460, 30.461689002093972809169984370455, 32.02410080610613396822657876442, 33.294108196740585695994661467777