L(s) = 1 | + (0.555 − 0.831i)3-s + (−0.195 + 0.980i)5-s + (0.382 − 0.923i)7-s + (−0.382 − 0.923i)9-s + (0.831 − 0.555i)11-s + (0.195 + 0.980i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.980 + 0.195i)19-s + (−0.555 − 0.831i)21-s + (0.923 − 0.382i)23-s + (−0.923 − 0.382i)25-s + (−0.980 − 0.195i)27-s + (−0.831 − 0.555i)29-s + i·31-s + ⋯ |
L(s) = 1 | + (0.555 − 0.831i)3-s + (−0.195 + 0.980i)5-s + (0.382 − 0.923i)7-s + (−0.382 − 0.923i)9-s + (0.831 − 0.555i)11-s + (0.195 + 0.980i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.980 + 0.195i)19-s + (−0.555 − 0.831i)21-s + (0.923 − 0.382i)23-s + (−0.923 − 0.382i)25-s + (−0.980 − 0.195i)27-s + (−0.831 − 0.555i)29-s + i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.197806620 - 0.4620449997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.197806620 - 0.4620449997i\) |
\(L(1)\) |
\(\approx\) |
\(1.202096253 - 0.2852291523i\) |
\(L(1)\) |
\(\approx\) |
\(1.202096253 - 0.2852291523i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.555 - 0.831i)T \) |
| 5 | \( 1 + (-0.195 + 0.980i)T \) |
| 7 | \( 1 + (0.382 - 0.923i)T \) |
| 11 | \( 1 + (0.831 - 0.555i)T \) |
| 13 | \( 1 + (0.195 + 0.980i)T \) |
| 17 | \( 1 + (0.707 - 0.707i)T \) |
| 19 | \( 1 + (-0.980 + 0.195i)T \) |
| 23 | \( 1 + (0.923 - 0.382i)T \) |
| 29 | \( 1 + (-0.831 - 0.555i)T \) |
| 31 | \( 1 + iT \) |
| 37 | \( 1 + (0.980 + 0.195i)T \) |
| 41 | \( 1 + (-0.923 + 0.382i)T \) |
| 43 | \( 1 + (0.555 + 0.831i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.831 + 0.555i)T \) |
| 59 | \( 1 + (0.195 - 0.980i)T \) |
| 61 | \( 1 + (-0.555 + 0.831i)T \) |
| 67 | \( 1 + (-0.555 + 0.831i)T \) |
| 71 | \( 1 + (-0.382 + 0.923i)T \) |
| 73 | \( 1 + (0.382 + 0.923i)T \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 + (0.980 - 0.195i)T \) |
| 89 | \( 1 + (0.923 + 0.382i)T \) |
| 97 | \( 1 + iT \) |
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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
−28.461463648413178508036272341571, −27.73035137398556643613163842398, −27.34999250890798027371745206861, −25.56896379366931097629364478899, −25.23202354749974399795352156966, −24.06586459756565608443229507748, −22.71714812645858100098839264806, −21.64040047253024834078373742592, −20.82800356578739309619800715459, −19.975086170718308419729053463916, −18.98906357044461270237488935893, −17.382591245297233596401975991687, −16.57239128807697120106584884873, −15.18565340227105819944218516526, −14.91121689350982518051385341707, −13.22091168685038300270266071472, −12.232816542875914465083257433162, −10.95731794958018390607540675000, −9.55569043883925440225346876890, −8.75770443809423603389187780515, −7.859590566880932829090742498803, −5.744167003935538363361460654290, −4.74611296960581491549128962541, −3.541724851697954470605827248883, −1.86571006722425331926189975166,
1.40615458747066348951707210442, 2.988183336231101290169476202095, 4.13577637136009243721556961790, 6.34417687794580114112361703055, 7.07223439849927364303229973558, 8.13440786544419540012568730224, 9.451385715284216720990715367353, 10.96082409703729294738904412127, 11.7794874940000526375893805006, 13.28091701730342714300558898452, 14.289579708513733609847637129816, 14.69413833412592146006111737299, 16.50047588667096818904431720972, 17.54465631740588006166019898088, 18.79383048366331008196484769093, 19.23790789345696482239533006078, 20.44882078278122416806256879866, 21.52478608696412270081183317240, 22.98786496348618945933625567014, 23.57204803235639706617769056184, 24.70443152308035266523402513913, 25.710297011289610423663477187198, 26.64266688866419460213709286574, 27.33400661926823284123905399383, 29.05922440786221996891876679390