L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.809 − 0.587i)3-s + (0.809 − 0.587i)4-s + (0.951 + 0.309i)5-s + (−0.951 − 0.309i)6-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + 10-s − 12-s + (−0.809 − 0.587i)14-s + (−0.587 − 0.809i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.587 + 0.809i)18-s + (−0.587 + 0.809i)19-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.809 − 0.587i)3-s + (0.809 − 0.587i)4-s + (0.951 + 0.309i)5-s + (−0.951 − 0.309i)6-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + 10-s − 12-s + (−0.809 − 0.587i)14-s + (−0.587 − 0.809i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.587 + 0.809i)18-s + (−0.587 + 0.809i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.291608998 - 0.9880279079i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.291608998 - 0.9880279079i\) |
\(L(1)\) |
\(\approx\) |
\(1.378634663 - 0.6354035006i\) |
\(L(1)\) |
\(\approx\) |
\(1.378634663 - 0.6354035006i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.951 + 0.309i)T \) |
| 7 | \( 1 + (-0.587 - 0.809i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.587 + 0.809i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.951 + 0.309i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (-0.587 + 0.809i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.587 + 0.809i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.587 + 0.809i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.951 + 0.309i)T \) |
| 73 | \( 1 + (-0.587 - 0.809i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (-0.951 + 0.309i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.677734426796633893161738223436, −27.830642419226645763310535683126, −26.110010942343611638215236537568, −25.58073686240540499766706129385, −24.35842740594771159475415965032, −23.52264690368291394756559074446, −22.30510002751261496611695962532, −21.75344558629007537381775222066, −21.136971998025033870287087681090, −19.82597405191523916088303574174, −18.11062989493251232877461769513, −17.129153490891322687298771447382, −16.26990583492391530275288926566, −15.379818544815872383933436586524, −14.32308128196753380130146969061, −12.900933203742137385135642845958, −12.36197748528637115262996095083, −11.03868044429100842563961820758, −9.91147253353423120399064553449, −8.65776857337058701042652086326, −6.66532139098793280263084536691, −5.890579473062477910082184095752, −5.091456968762139320280399159395, −3.738648057541985771248466811348, −2.1669266720892368256996871806,
1.36577911433103401255319718282, 2.755138030557480367474667133349, 4.38956017317881601594557590332, 5.73666038462512019932016398220, 6.457922893905631640152473803884, 7.47272510466119061703863225308, 9.890024411319433055082859363467, 10.54063425714662562058869080546, 11.74054909151327844618592227263, 12.8199462146849414785403040117, 13.60598328960670888951062393567, 14.39211742033396541010152148638, 16.09212749826268045507253942318, 16.85825517063730625641047650744, 18.105977453109207435261057740495, 19.11211964400006590676573406618, 20.27155920208145116503275723312, 21.39443459545755262221414038153, 22.35270560439843469698494335073, 23.00613548238853910778951476231, 23.87579021270069303139145854694, 24.99658818460525858823157860385, 25.71461204208448436421910083926, 27.33486427964164977693293316462, 28.667345774488559969858586112990