L(s) = 1 | + (−0.951 − 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)6-s + (−0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s − i·12-s + (0.951 + 0.309i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (0.951 − 0.309i)17-s + (0.587 − 0.809i)18-s + (0.809 − 0.587i)19-s + 21-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)6-s + (−0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s − i·12-s + (0.951 + 0.309i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (0.951 − 0.309i)17-s + (0.587 − 0.809i)18-s + (0.809 − 0.587i)19-s + 21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7484616253 + 0.01907876892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7484616253 + 0.01907876892i\) |
\(L(1)\) |
\(\approx\) |
\(0.6163827331 - 0.08905143846i\) |
\(L(1)\) |
\(\approx\) |
\(0.6163827331 - 0.08905143846i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + (-0.587 + 0.809i)T \) |
| 13 | \( 1 + (0.951 + 0.309i)T \) |
| 17 | \( 1 + (0.951 - 0.309i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.587 + 0.809i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.587 + 0.809i)T \) |
| 53 | \( 1 + (0.951 + 0.309i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.587 - 0.809i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.951 + 0.309i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.951 - 0.309i)T \) |
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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
−32.95517450456803278474476783513, −32.37692539609439443280445403310, −30.24175240052944136471083630599, −29.04150426298800103160872111315, −28.25038529388461480086611300741, −27.09850521159285965606026794306, −26.33566825469337702749443292562, −25.233502206334605995320724834519, −23.58324936015139494024388294595, −22.78397007363319867048723357341, −21.03629092509502655414439549169, −20.16029939582020531707894780850, −18.70186310325976238548615688377, −17.43678902622106882327585227717, −16.44588239322257358693925769518, −15.6970558631187669245989963398, −14.18822097166114706041453106269, −12.09498402689116765413278702499, −10.60855650223915854769834485611, −10.00036964853372296332272426493, −8.50386718641188739054797759899, −6.823234583240222076536071693186, −5.5729348910396304893484854153, −3.5488411235447611735382960501, −0.75035636352011742738407224993,
1.1978145755402085723541765587, 2.93722983947841912891144227221, 5.77085977827002237771890230046, 7.0004695767335708138007093085, 8.373409640451886274533744488885, 9.75424733910661939123376116100, 11.335953859769432083167515867470, 12.157353286162630709694478351598, 13.457015210653557414569908073601, 15.669664494511695980610517818492, 16.66098512016301513367476001051, 18.00434273949785516673475083302, 18.71950409935262520412071750276, 19.71301543100906993127650241944, 21.26154439299629059532543989221, 22.498938391494291930039986530616, 23.91976923851351165788273766669, 25.17179634530296635255112126318, 25.87942696241736325185009644285, 27.586077880047096001629881712535, 28.42830150328469290109385116297, 29.240220294305608870146623839687, 30.30737192079687353608167226415, 31.371503130589660286278833066565, 33.24621674652445905135298541573