L(s) = 1 | + (0.980 − 0.195i)5-s + (0.382 + 0.923i)7-s + (0.555 − 0.831i)11-s + (0.980 + 0.195i)13-s + (−0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (−0.923 − 0.382i)23-s + (0.923 − 0.382i)25-s + (0.555 + 0.831i)29-s + i·31-s + (0.555 + 0.831i)35-s + (−0.195 − 0.980i)37-s + (−0.923 − 0.382i)41-s + (0.831 + 0.555i)43-s + (−0.707 − 0.707i)47-s + ⋯ |
L(s) = 1 | + (0.980 − 0.195i)5-s + (0.382 + 0.923i)7-s + (0.555 − 0.831i)11-s + (0.980 + 0.195i)13-s + (−0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (−0.923 − 0.382i)23-s + (0.923 − 0.382i)25-s + (0.555 + 0.831i)29-s + i·31-s + (0.555 + 0.831i)35-s + (−0.195 − 0.980i)37-s + (−0.923 − 0.382i)41-s + (0.831 + 0.555i)43-s + (−0.707 − 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.677035102 + 0.1237055409i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.677035102 + 0.1237055409i\) |
\(L(1)\) |
\(\approx\) |
\(1.334552535 + 0.04939433649i\) |
\(L(1)\) |
\(\approx\) |
\(1.334552535 + 0.04939433649i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.980 - 0.195i)T \) |
| 7 | \( 1 + (0.382 + 0.923i)T \) |
| 11 | \( 1 + (0.555 - 0.831i)T \) |
| 13 | \( 1 + (0.980 + 0.195i)T \) |
| 17 | \( 1 + (-0.707 - 0.707i)T \) |
| 19 | \( 1 + (-0.195 + 0.980i)T \) |
| 23 | \( 1 + (-0.923 - 0.382i)T \) |
| 29 | \( 1 + (0.555 + 0.831i)T \) |
| 31 | \( 1 + iT \) |
| 37 | \( 1 + (-0.195 - 0.980i)T \) |
| 41 | \( 1 + (-0.923 - 0.382i)T \) |
| 43 | \( 1 + (0.831 + 0.555i)T \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.555 - 0.831i)T \) |
| 59 | \( 1 + (0.980 - 0.195i)T \) |
| 61 | \( 1 + (0.831 - 0.555i)T \) |
| 67 | \( 1 + (-0.831 + 0.555i)T \) |
| 71 | \( 1 + (0.382 + 0.923i)T \) |
| 73 | \( 1 + (-0.382 + 0.923i)T \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 + (-0.195 + 0.980i)T \) |
| 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
−24.41630259923152439557212741194, −23.70656377773130412775351432678, −22.65867895678777665670000685846, −21.946361290457041655431605942878, −20.91254312681455083287550023639, −20.282059723180731424447195145722, −19.34777173954271994870958744439, −18.01418755252915464737739408228, −17.54553347168992140849983859967, −16.8317279374829212976299356655, −15.52426160176351401699909286878, −14.675509641287711075733836512676, −13.556365290723491964190688081183, −13.27988513671878082968765476204, −11.81237667427665818417359535819, −10.79400908490481587520589379393, −10.06978749246160415364688919814, −9.09266753027972009212405544718, −7.97176481649235337418117810461, −6.778401505763132776670157506170, −6.10275829290797136565161669525, −4.72362070138373878669149753554, −3.830327521430855064158027111328, −2.28479381134179696274472999858, −1.28251711843656962119282885676,
1.37912851286621529313223745649, 2.37450801964620767045326140, 3.70216838237664380365487524068, 5.105513448110488967105798350799, 5.932502172650638504492895285720, 6.71195128991265579424087095141, 8.511095349642666350459667877305, 8.779818576359700817303872550046, 9.97370559279492480457772091031, 11.02794519085384374223586610297, 11.95022259428753917210655603400, 12.93399897332746300087783131574, 14.006754974083798226626249025410, 14.46464020333052419518549545681, 15.92667280882835303990529485207, 16.43556985938528152186989852150, 17.76745610442732732620746318537, 18.202920721798044140795793429591, 19.15321930915499247004115076994, 20.36310600697115906716514644451, 21.18843864946781327765989061904, 21.80230035425462446900643631895, 22.606440890821668023221602407200, 23.84116193095430658060998487236, 24.75475203910188224588059205377