L(s) = 1 | + (0.923 − 0.382i)3-s + (0.831 − 0.555i)5-s + (−0.831 − 0.555i)7-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.555 − 0.831i)13-s + (0.555 − 0.831i)15-s + (−0.555 − 0.831i)17-s + (0.831 − 0.555i)19-s + (−0.980 − 0.195i)21-s + (−0.195 + 0.980i)23-s + (0.382 − 0.923i)25-s + (0.382 − 0.923i)27-s + (0.195 − 0.980i)29-s + (0.382 + 0.923i)31-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)3-s + (0.831 − 0.555i)5-s + (−0.831 − 0.555i)7-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.555 − 0.831i)13-s + (0.555 − 0.831i)15-s + (−0.555 − 0.831i)17-s + (0.831 − 0.555i)19-s + (−0.980 − 0.195i)21-s + (−0.195 + 0.980i)23-s + (0.382 − 0.923i)25-s + (0.382 − 0.923i)27-s + (0.195 − 0.980i)29-s + (0.382 + 0.923i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.217857488 - 1.191947853i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.217857488 - 1.191947853i\) |
\(L(1)\) |
\(\approx\) |
\(1.283683115 - 0.5382743795i\) |
\(L(1)\) |
\(\approx\) |
\(1.283683115 - 0.5382743795i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 97 | \( 1 \) |
good | 3 | \( 1 + (0.923 - 0.382i)T \) |
| 5 | \( 1 + (0.831 - 0.555i)T \) |
| 7 | \( 1 + (-0.831 - 0.555i)T \) |
| 11 | \( 1 + (-0.923 + 0.382i)T \) |
| 13 | \( 1 + (-0.555 - 0.831i)T \) |
| 17 | \( 1 + (-0.555 - 0.831i)T \) |
| 19 | \( 1 + (0.831 - 0.555i)T \) |
| 23 | \( 1 + (-0.195 + 0.980i)T \) |
| 29 | \( 1 + (0.195 - 0.980i)T \) |
| 31 | \( 1 + (0.382 + 0.923i)T \) |
| 37 | \( 1 + (-0.980 + 0.195i)T \) |
| 41 | \( 1 + (0.980 + 0.195i)T \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.923 + 0.382i)T \) |
| 59 | \( 1 + (0.195 + 0.980i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.555 + 0.831i)T \) |
| 71 | \( 1 + (-0.980 + 0.195i)T \) |
| 73 | \( 1 + (-0.707 + 0.707i)T \) |
| 79 | \( 1 + (-0.382 - 0.923i)T \) |
| 83 | \( 1 + (0.831 - 0.555i)T \) |
| 89 | \( 1 + (0.923 - 0.382i)T \) |
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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
−24.7567731268550604373198097848, −24.151341560727458187269927569836, −22.58999858770022633731168767489, −21.99769840950611665663584314967, −21.26689903379134309224915567676, −20.4842612212151282190175515688, −19.23614101161576788378742949214, −18.83449088648478033033660624970, −17.8563214723721613013948474479, −16.52246239575604063149702203326, −15.834954320496938174844709101783, −14.82630526873177977776202446015, −14.09569694877596571600678746746, −13.25271263377509224102773477997, −12.416632324964754335965542532890, −10.845934386410391423028929458345, −10.066757285820347574479914126504, −9.32696634560008584530066968842, −8.44697764802639153156726360096, −7.220257682247565469363511255750, −6.21063590961981249763846792857, −5.13502868851051143730327243669, −3.73741369792109794012313420683, −2.70358917707031462455422725142, −2.03654260248727031118652848121,
0.89441858193624736406691630209, 2.375191401051811701843573068106, 3.09978556172129298016559504855, 4.564011458155837267974139303472, 5.64997566232797201849514058801, 6.962630479945424793245304977528, 7.65683911643217829645950697123, 8.85528011236537578557745055213, 9.75716972924241625697653242603, 10.23515624768697108965547924619, 11.97280176269356505777490973482, 13.11453934290218813372741514968, 13.31349144570976680352747938321, 14.23737492336821154781416673743, 15.546253455085362623844451987344, 16.08379222651129734378563124408, 17.607650478934760914603372659021, 17.89295258430761068655247699530, 19.241229856648095050478326016899, 20.00527866046174254109646246481, 20.58848421169780509695890448789, 21.46095849069221379925353694626, 22.565979557135728994011178830829, 23.484225579052901504749010625268, 24.62879210782882413171717422250