L(s) = 1 | + (−0.984 − 0.173i)3-s + (0.866 − 0.5i)7-s + (0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.342 − 0.939i)17-s + (−0.939 + 0.342i)21-s + (−0.642 − 0.766i)23-s + (−0.866 − 0.5i)27-s + (0.939 + 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.642 + 0.766i)33-s − i·37-s − 39-s + (−0.173 + 0.984i)41-s + ⋯ |
L(s) = 1 | + (−0.984 − 0.173i)3-s + (0.866 − 0.5i)7-s + (0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.342 − 0.939i)17-s + (−0.939 + 0.342i)21-s + (−0.642 − 0.766i)23-s + (−0.866 − 0.5i)27-s + (0.939 + 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.642 + 0.766i)33-s − i·37-s − 39-s + (−0.173 + 0.984i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.002674537 - 0.6630913597i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.002674537 - 0.6630913597i\) |
\(L(1)\) |
\(\approx\) |
\(0.9188633732 - 0.2324428548i\) |
\(L(1)\) |
\(\approx\) |
\(0.9188633732 - 0.2324428548i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.984 - 0.173i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.342 - 0.939i)T \) |
| 23 | \( 1 + (-0.642 - 0.766i)T \) |
| 29 | \( 1 + (0.939 + 0.342i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.642 + 0.766i)T \) |
| 47 | \( 1 + (0.342 - 0.939i)T \) |
| 53 | \( 1 + (-0.642 - 0.766i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (-0.766 + 0.642i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (-0.766 - 0.642i)T \) |
| 73 | \( 1 + (-0.984 - 0.173i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.342 + 0.939i)T \) |
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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
−22.51192544445938326940694639322, −21.79217123320085150654337729169, −21.112742548737483334864039217200, −20.33985389448912792233393869207, −19.17491793129650829886231431685, −18.348692111447890134819923477999, −17.474496608136799314551665450585, −17.27657777956227689210861062121, −15.9098451753869759454309085938, −15.44817655213533407155733269816, −14.52332811787704395316048886706, −13.476113611074753954385841187713, −12.421270217191723995093402002184, −11.809746805224166748903780661594, −11.06208391906885981021560500933, −10.25877718956784677085176643036, −9.26305525429913558580571350989, −8.30669032225067432425587218708, −7.26874589519781680811913973121, −6.23014086927650299981594324579, −5.60544645611954116397956998303, −4.47867591014733867344901250328, −3.898348286830783427142910446124, −2.09901078507284976056774645352, −1.25836356061658474180637060023,
0.76323480452570374323434214462, 1.614978483449765021582312044313, 3.20241309665892877899728026535, 4.40789815946397385644389286221, 5.06259072178654245104245821156, 6.22110465226219435305838363090, 6.77436435661985115376745700091, 7.979867144718211019319218396756, 8.68528322603829724123038366889, 10.05368643678848556360124483066, 10.80598357332876532407843224593, 11.5022299954272612806849588377, 12.0917425766141294994573268126, 13.39639416517192403712842714622, 13.84321813418207440883780416482, 14.93322257187217957267992729516, 16.177969481093299761162476616214, 16.418070808264133306407273410300, 17.57115781722170469383318428051, 18.05869803833861536497106835165, 18.800977881971282297090837057281, 19.89892499725374780011473213759, 20.7616683614443650116616211796, 21.55428516959674702080103573901, 22.282249028357560056383552914071