L(s) = 1 | + (−0.771 + 0.636i)2-s + (0.817 + 0.575i)3-s + (0.190 − 0.981i)4-s + (−0.973 − 0.227i)5-s + (−0.997 + 0.0765i)6-s + (0.720 + 0.693i)7-s + (0.477 + 0.878i)8-s + (0.338 + 0.941i)9-s + (0.896 − 0.443i)10-s + (−0.665 + 0.746i)11-s + (0.720 − 0.693i)12-s + (0.953 + 0.301i)13-s + (−0.997 − 0.0765i)14-s + (−0.665 − 0.746i)15-s + (−0.927 − 0.373i)16-s + (0.606 − 0.795i)17-s + ⋯ |
L(s) = 1 | + (−0.771 + 0.636i)2-s + (0.817 + 0.575i)3-s + (0.190 − 0.981i)4-s + (−0.973 − 0.227i)5-s + (−0.997 + 0.0765i)6-s + (0.720 + 0.693i)7-s + (0.477 + 0.878i)8-s + (0.338 + 0.941i)9-s + (0.896 − 0.443i)10-s + (−0.665 + 0.746i)11-s + (0.720 − 0.693i)12-s + (0.953 + 0.301i)13-s + (−0.997 − 0.0765i)14-s + (−0.665 − 0.746i)15-s + (−0.927 − 0.373i)16-s + (0.606 − 0.795i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0645 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0645 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5404492542 + 0.5765310567i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5404492542 + 0.5765310567i\) |
\(L(1)\) |
\(\approx\) |
\(0.7351142803 + 0.4339409430i\) |
\(L(1)\) |
\(\approx\) |
\(0.7351142803 + 0.4339409430i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (-0.771 + 0.636i)T \) |
| 3 | \( 1 + (0.817 + 0.575i)T \) |
| 5 | \( 1 + (-0.973 - 0.227i)T \) |
| 7 | \( 1 + (0.720 + 0.693i)T \) |
| 11 | \( 1 + (-0.665 + 0.746i)T \) |
| 13 | \( 1 + (0.953 + 0.301i)T \) |
| 17 | \( 1 + (0.606 - 0.795i)T \) |
| 19 | \( 1 + (-0.543 + 0.839i)T \) |
| 23 | \( 1 + (-0.859 + 0.511i)T \) |
| 29 | \( 1 + (-0.409 - 0.912i)T \) |
| 31 | \( 1 + (0.477 - 0.878i)T \) |
| 37 | \( 1 + (0.338 - 0.941i)T \) |
| 41 | \( 1 + (-0.771 - 0.636i)T \) |
| 43 | \( 1 + (-0.264 - 0.964i)T \) |
| 47 | \( 1 + (0.988 - 0.152i)T \) |
| 53 | \( 1 + (0.988 + 0.152i)T \) |
| 59 | \( 1 + (-0.114 + 0.993i)T \) |
| 61 | \( 1 + (0.0383 - 0.999i)T \) |
| 67 | \( 1 + (-0.927 - 0.373i)T \) |
| 71 | \( 1 + (0.720 - 0.693i)T \) |
| 73 | \( 1 + (0.896 - 0.443i)T \) |
| 79 | \( 1 + (0.190 - 0.981i)T \) |
| 89 | \( 1 + (-0.997 + 0.0765i)T \) |
| 97 | \( 1 + (-0.997 - 0.0765i)T \) |
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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
−30.34349543716782678497302966952, −29.94849949264037791352566305117, −28.3819117634213436704788862182, −27.305766718189313065792614232593, −26.43712592615040918454516405577, −25.71028564881729022314322260692, −24.14564484486100646178896302466, −23.4044481378032851689027533005, −21.53125590678585656402250896372, −20.49924731622918439394024557052, −19.74017222806127386031568267923, −18.735300321954127572474237050509, −17.962236970453601189037212562323, −16.44668768364700217602060719663, −15.175489380129390896535918615629, −13.73526515774019187062021404123, −12.6267263215043341262858603326, −11.31039855846221792447250791875, −10.39576994221521739382051682563, −8.39826087608884361996726682740, −8.15992239695028848062290236993, −6.85084500881427559118740021086, −4.03795885934323933340347800975, −2.973686664479458292313737661155, −1.17190879687712240283423513421,
2.07848542871376719742207135157, 4.14126108379850092447182184289, 5.47639655605151990412474826587, 7.57963645016326257254526228638, 8.20971129741316575178340300702, 9.27910725509577834957624188582, 10.57720593668821375707684914795, 11.87827531391496604290957473788, 13.83491199022743676231646596021, 15.097242420613877348156061245954, 15.58402235977571685695054787282, 16.64022264872084879005427504653, 18.32065058073489683837575009459, 19.016526820168596982346764192870, 20.35375027940794017371277818812, 20.979233564380558902751089819039, 22.87437701091842031371038281364, 23.922119477579706650725252972204, 25.04279197707645886509001229703, 25.86354367415171597887746299165, 26.95599152008897346255270399909, 27.79467584797900263414094622658, 28.34883429329039807947639783260, 30.34993162123360137644741856881, 31.51259582748972382207341894966