L(s) = 1 | + (0.953 + 0.301i)2-s + (−0.997 − 0.0765i)3-s + (0.817 + 0.575i)4-s + (−0.409 + 0.912i)5-s + (−0.927 − 0.373i)6-s + (−0.771 + 0.636i)7-s + (0.606 + 0.795i)8-s + (0.988 + 0.152i)9-s + (−0.665 + 0.746i)10-s + (0.477 + 0.878i)11-s + (−0.771 − 0.636i)12-s + (0.0383 − 0.999i)13-s + (−0.927 + 0.373i)14-s + (0.477 − 0.878i)15-s + (0.338 + 0.941i)16-s + (−0.114 − 0.993i)17-s + ⋯ |
L(s) = 1 | + (0.953 + 0.301i)2-s + (−0.997 − 0.0765i)3-s + (0.817 + 0.575i)4-s + (−0.409 + 0.912i)5-s + (−0.927 − 0.373i)6-s + (−0.771 + 0.636i)7-s + (0.606 + 0.795i)8-s + (0.988 + 0.152i)9-s + (−0.665 + 0.746i)10-s + (0.477 + 0.878i)11-s + (−0.771 − 0.636i)12-s + (0.0383 − 0.999i)13-s + (−0.927 + 0.373i)14-s + (0.477 − 0.878i)15-s + (0.338 + 0.941i)16-s + (−0.114 − 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8399871680 + 0.7534274722i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8399871680 + 0.7534274722i\) |
\(L(1)\) |
\(\approx\) |
\(1.084830588 + 0.5151792589i\) |
\(L(1)\) |
\(\approx\) |
\(1.084830588 + 0.5151792589i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (0.953 + 0.301i)T \) |
| 3 | \( 1 + (-0.997 - 0.0765i)T \) |
| 5 | \( 1 + (-0.409 + 0.912i)T \) |
| 7 | \( 1 + (-0.771 + 0.636i)T \) |
| 11 | \( 1 + (0.477 + 0.878i)T \) |
| 13 | \( 1 + (0.0383 - 0.999i)T \) |
| 17 | \( 1 + (-0.114 - 0.993i)T \) |
| 19 | \( 1 + (-0.264 + 0.964i)T \) |
| 23 | \( 1 + (0.896 - 0.443i)T \) |
| 29 | \( 1 + (-0.859 - 0.511i)T \) |
| 31 | \( 1 + (0.606 - 0.795i)T \) |
| 37 | \( 1 + (0.988 - 0.152i)T \) |
| 41 | \( 1 + (0.953 - 0.301i)T \) |
| 43 | \( 1 + (-0.973 + 0.227i)T \) |
| 47 | \( 1 + (0.720 + 0.693i)T \) |
| 53 | \( 1 + (0.720 - 0.693i)T \) |
| 59 | \( 1 + (-0.543 - 0.839i)T \) |
| 61 | \( 1 + (0.190 + 0.981i)T \) |
| 67 | \( 1 + (0.338 + 0.941i)T \) |
| 71 | \( 1 + (-0.771 - 0.636i)T \) |
| 73 | \( 1 + (-0.665 + 0.746i)T \) |
| 79 | \( 1 + (0.817 + 0.575i)T \) |
| 89 | \( 1 + (-0.927 - 0.373i)T \) |
| 97 | \( 1 + (-0.927 + 0.373i)T \) |
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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
−30.54109047042829928781114729161, −29.46617235095767548585095116585, −28.73505042521013929334814663758, −27.90241475886044119696883269466, −26.50813278749146272870503713402, −24.78767686975454684133021252068, −23.74598003168081179310172968739, −23.36033135600789325045973425138, −21.980459317378360395924569986545, −21.297946919346620363185182779911, −19.83763574280088623114078917024, −19.075491026768949882535782157273, −16.92992717412535092052918438328, −16.446925043832014013469996780008, −15.314795964494617396435046402591, −13.55135177588856720571012367277, −12.76215679274967254416311980877, −11.64288040406584098528050462216, −10.764002564676658695421133505382, −9.2365540845023234351159385853, −7.04809489648262711127535761227, −6.021380841807035724645473282596, −4.65285392671030571821777117750, −3.7010378639217365164770206012, −1.16681688387130208026942130459,
2.62640004241850237489589569044, 4.12232556093047870072922901928, 5.63150432903936505928451646650, 6.5961241346014327100004416570, 7.5854179503104658616190213661, 9.95932141382335657338106187774, 11.249776845364752603156412824139, 12.1866415841278246846260718637, 13.09369992721623849392782097956, 14.81027750558449773889006111797, 15.543415209851775233070472891014, 16.641698592511709693427372447970, 17.93535160523014940391987296786, 19.093351392803320613287914476767, 20.59799960566801378957947800194, 22.03444950504501380259370922697, 22.830715038103940050709430570971, 22.9867022990939111779064076869, 24.69155692362513986116800928237, 25.44939406149688683167704264108, 26.87771750910400876346394960440, 28.10019777699950841651682539325, 29.34681339909836591582464624553, 30.05338894261507242592175564300, 31.098083312925532143422918435284