L(s) = 1 | + (−0.888 − 0.458i)2-s + (−0.959 + 0.281i)3-s + (0.580 + 0.814i)4-s + (−0.654 − 0.755i)5-s + (0.981 + 0.189i)6-s + (0.0475 + 0.998i)7-s + (−0.142 − 0.989i)8-s + (0.841 − 0.540i)9-s + (0.235 + 0.971i)10-s + (0.981 − 0.189i)11-s + (−0.786 − 0.618i)12-s + (0.928 − 0.371i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (−0.327 + 0.945i)16-s + (0.580 − 0.814i)17-s + ⋯ |
L(s) = 1 | + (−0.888 − 0.458i)2-s + (−0.959 + 0.281i)3-s + (0.580 + 0.814i)4-s + (−0.654 − 0.755i)5-s + (0.981 + 0.189i)6-s + (0.0475 + 0.998i)7-s + (−0.142 − 0.989i)8-s + (0.841 − 0.540i)9-s + (0.235 + 0.971i)10-s + (0.981 − 0.189i)11-s + (−0.786 − 0.618i)12-s + (0.928 − 0.371i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (−0.327 + 0.945i)16-s + (0.580 − 0.814i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4476095019 - 0.1201365145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4476095019 - 0.1201365145i\) |
\(L(1)\) |
\(\approx\) |
\(0.5423510219 - 0.09358056119i\) |
\(L(1)\) |
\(\approx\) |
\(0.5423510219 - 0.09358056119i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (-0.888 - 0.458i)T \) |
| 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.0475 + 0.998i)T \) |
| 11 | \( 1 + (0.981 - 0.189i)T \) |
| 13 | \( 1 + (0.928 - 0.371i)T \) |
| 17 | \( 1 + (0.580 - 0.814i)T \) |
| 19 | \( 1 + (0.0475 - 0.998i)T \) |
| 23 | \( 1 + (0.723 + 0.690i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.928 + 0.371i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.995 - 0.0950i)T \) |
| 43 | \( 1 + (0.415 + 0.909i)T \) |
| 47 | \( 1 + (0.235 - 0.971i)T \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.142 - 0.989i)T \) |
| 61 | \( 1 + (0.981 + 0.189i)T \) |
| 71 | \( 1 + (0.580 + 0.814i)T \) |
| 73 | \( 1 + (0.981 + 0.189i)T \) |
| 79 | \( 1 + (-0.786 - 0.618i)T \) |
| 83 | \( 1 + (-0.327 + 0.945i)T \) |
| 89 | \( 1 + (-0.959 - 0.281i)T \) |
| 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
−32.67546744738897062913095880002, −30.498938968312542280240251636219, −29.88314797607390580825018098995, −28.62915883510868055050324641601, −27.56319788401148992227999095951, −26.860472260773799329469822923331, −25.677675334745769871246666288095, −24.27749471205526149330704842719, −23.306475320072030770457017182745, −22.65287023591780614982027866014, −20.72031467741408827287106091475, −19.24149528885590259979516048654, −18.61638383297541808178923000242, −17.22339245962497513838374276470, −16.6112207821727810678268778395, −15.26876944860360647933959989055, −13.972427176011718566723256008958, −11.957715963136077650821040303745, −10.945087430718118045569730372, −10.09746421134974055212241306893, −8.11418599937185890644803537308, −6.967901683426904726580625417299, −6.15590436368281500966343579279, −4.069593453637532472664190925779, −1.28467241471813464839178007330,
1.11083343202195577276579051098, 3.54297348762982222466817064650, 5.267459972123692533635767224641, 6.90074790911222683352880692616, 8.59288972043066946275005767119, 9.475336159279833181076349694346, 11.20969354155861326455035451786, 11.79903084222927874260656321702, 12.852064443975353622090878802373, 15.45243877598710280508225033382, 16.17962746287633893007582454096, 17.26505513282326240584527461009, 18.34454714863245805110219352466, 19.45097742329842484192601704812, 20.75998473445905613113211797305, 21.709237725141650891684361319293, 22.91734205866315282571446456235, 24.36721221923527848961241317978, 25.32299983148287022394431876389, 26.98367854189166641770042641485, 27.89556659030022287831871655033, 28.19529506104407408670767129923, 29.43494590962517682776110222545, 30.5735969254018922718377552739, 31.8957994119506463419416819895