L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.800 − 0.599i)3-s + (−0.142 − 0.989i)4-s + (0.540 − 0.841i)5-s + (−0.0713 + 0.997i)6-s + (−0.212 − 0.977i)7-s + (0.841 + 0.540i)8-s + (0.281 − 0.959i)9-s + (0.281 + 0.959i)10-s + (−0.841 + 0.540i)11-s + (−0.707 − 0.707i)12-s + (−0.599 − 0.800i)13-s + (0.877 + 0.479i)14-s + (−0.0713 − 0.997i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 0.654i)17-s + ⋯ |
L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.800 − 0.599i)3-s + (−0.142 − 0.989i)4-s + (0.540 − 0.841i)5-s + (−0.0713 + 0.997i)6-s + (−0.212 − 0.977i)7-s + (0.841 + 0.540i)8-s + (0.281 − 0.959i)9-s + (0.281 + 0.959i)10-s + (−0.841 + 0.540i)11-s + (−0.707 − 0.707i)12-s + (−0.599 − 0.800i)13-s + (0.877 + 0.479i)14-s + (−0.0713 − 0.997i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 0.654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8035023608 - 0.9660466464i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8035023608 - 0.9660466464i\) |
\(L(1)\) |
\(\approx\) |
\(0.9148053658 - 0.2780762479i\) |
\(L(1)\) |
\(\approx\) |
\(0.9148053658 - 0.2780762479i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.800 - 0.599i)T \) |
| 5 | \( 1 + (0.540 - 0.841i)T \) |
| 7 | \( 1 + (-0.212 - 0.977i)T \) |
| 11 | \( 1 + (-0.841 + 0.540i)T \) |
| 13 | \( 1 + (-0.599 - 0.800i)T \) |
| 17 | \( 1 + (-0.755 + 0.654i)T \) |
| 19 | \( 1 + (-0.877 + 0.479i)T \) |
| 23 | \( 1 + (0.479 + 0.877i)T \) |
| 29 | \( 1 + (0.977 - 0.212i)T \) |
| 31 | \( 1 + (0.479 - 0.877i)T \) |
| 37 | \( 1 + (0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.599 - 0.800i)T \) |
| 43 | \( 1 + (0.977 + 0.212i)T \) |
| 47 | \( 1 + (-0.989 + 0.142i)T \) |
| 53 | \( 1 + (0.989 + 0.142i)T \) |
| 59 | \( 1 + (-0.800 - 0.599i)T \) |
| 61 | \( 1 + (0.936 - 0.349i)T \) |
| 67 | \( 1 + (-0.142 + 0.989i)T \) |
| 71 | \( 1 + (-0.540 - 0.841i)T \) |
| 73 | \( 1 + (0.959 - 0.281i)T \) |
| 79 | \( 1 + (-0.281 - 0.959i)T \) |
| 83 | \( 1 + (0.0713 - 0.997i)T \) |
| 97 | \( 1 + (0.841 + 0.540i)T \) |
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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
−30.68552790821105923265129854375, −29.29313061103912045068209427229, −28.590723897978487685514105806557, −27.19343227216274349525933915900, −26.46306276687793545210069242654, −25.69804914285747585818444029834, −24.7192592008210579555214951175, −22.55519049192061944643047531434, −21.523996307087088130560817482413, −21.24537615605069912796660349045, −19.654016850652107870849315386565, −18.86859772354948488037246410355, −17.99623754356996427037171185433, −16.451523613850737211636325788795, −15.32703633007801363468260106552, −14.03780398013912990744513762083, −12.91380376169940591278784575652, −11.304486873877405721125837519488, −10.30719812210233918825334928841, −9.2791180083174310178241827152, −8.39085934183671507471711460869, −6.79192552809507817670030348404, −4.69883334568798661375774321044, −2.830564871206753907732120396419, −2.40355637160477953677864645242,
0.62755133181194289322458466678, 2.12447549035521357354775518880, 4.47055827734584369235429098984, 6.07076147473105896187849635021, 7.42543414458771602209818969114, 8.24893804505182216261071779820, 9.506865278147114679650880787547, 10.433820979820747718328110304088, 12.79073328926913945094833034169, 13.45488733197152854945482883880, 14.71183138259170256048274305883, 15.81285231768764980111370864688, 17.27850603937110849983748446927, 17.74118975136775136219014226507, 19.30608262990898058486319706340, 20.031981237648302872729136075120, 21.00940902488127564659544805442, 23.09254377237762692188060589823, 23.935177264429123217806149224362, 24.86303148959643885429632853182, 25.704105524793282767674583354824, 26.50182679973792978772256605666, 27.695144840037103799696062206294, 29.010180325087663929581901506222, 29.66942115219438562922843619232