L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + 5-s − 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.309 + 0.951i)11-s + (0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + 5-s − 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.309 + 0.951i)11-s + (0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.263485602 + 0.6861757113i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.263485602 + 0.6861757113i\) |
\(L(1)\) |
\(\approx\) |
\(1.053460717 + 0.4046206362i\) |
\(L(1)\) |
\(\approx\) |
\(1.053460717 + 0.4046206362i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + (0.809 - 0.587i)T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.809 - 0.587i)T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−36.687546405675858072788100419069, −35.33161020921246885381153605777, −34.33112090214537823899856566049, −32.43310552336537297609135857137, −31.060319053779928034103084912540, −30.005394132694299012226287795271, −29.01951652168037335796314888442, −27.73608994586007887660688993422, −26.07897950654395946136932602383, −25.41840823263600356123861538673, −24.241850570423611535179623319689, −21.70233834070759406714664924548, −21.01510923350339695046907606948, −19.49171642349767974418796295770, −18.39551380196769157359724300330, −17.51633508992540963445166005648, −15.51449148904393402888051974522, −13.65561019591491120204956644703, −12.5661378758048446697706336889, −10.76140316235983546624840629694, −9.06655662913371574782475996910, −8.2665872422878573757429102791, −6.21471622633550805039157433271, −3.02314007770844106983583767971, −1.63373215918685350596076640681,
1.9575110911927671996819340224, 4.70537739853280268441677070753, 6.78587563857315644526855742999, 8.382017691832167994469099926492, 9.743978946147242402454810485946, 10.62685027631303779612388912885, 13.632252253007252212759739530260, 14.54112386648540209978080324748, 16.05896232245202353545686819417, 17.26676079122562009841168679365, 18.582300535998321819840690501829, 20.24662057477490309204052670546, 20.98315621033433075010335712111, 23.01611571659203062093857124786, 24.65977506247253393639896099802, 25.72776733429796362458188710719, 26.46002958084717865813369353190, 27.727513912860947960025559247302, 29.03603797667819291572472798408, 30.540233406937276302805905379717, 32.26341591888114120131762511493, 33.36540498809011424117486673539, 33.7627320533376208152470772283, 36.18254404947046433419816951247, 36.34586104068203256795808869184