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
−22.282249028357560056383552914071, −21.55428516959674702080103573901, −20.7616683614443650116616211796, −19.89892499725374780011473213759, −18.800977881971282297090837057281, −18.05869803833861536497106835165, −17.57115781722170469383318428051, −16.418070808264133306407273410300, −16.177969481093299761162476616214, −14.93322257187217957267992729516, −13.84321813418207440883780416482, −13.39639416517192403712842714622, −12.0917425766141294994573268126, −11.5022299954272612806849588377, −10.80598357332876532407843224593, −10.05368643678848556360124483066, −8.68528322603829724123038366889, −7.979867144718211019319218396756, −6.77436435661985115376745700091, −6.22110465226219435305838363090, −5.06259072178654245104245821156, −4.40789815946397385644389286221, −3.20241309665892877899728026535, −1.614978483449765021582312044313, −0.76323480452570374323434214462,
1.25836356061658474180637060023, 2.09901078507284976056774645352, 3.898348286830783427142910446124, 4.47867591014733867344901250328, 5.60544645611954116397956998303, 6.23014086927650299981594324579, 7.26874589519781680811913973121, 8.30669032225067432425587218708, 9.26305525429913558580571350989, 10.25877718956784677085176643036, 11.06208391906885981021560500933, 11.809746805224166748903780661594, 12.421270217191723995093402002184, 13.476113611074753954385841187713, 14.52332811787704395316048886706, 15.44817655213533407155733269816, 15.9098451753869759454309085938, 17.27657777956227689210861062121, 17.474496608136799314551665450585, 18.348692111447890134819923477999, 19.17491793129650829886231431685, 20.33985389448912792233393869207, 21.112742548737483334864039217200, 21.79217123320085150654337729169, 22.51192544445938326940694639322