| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1123 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1123 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6220641536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6220641536\) |
| \(L(1)\) |
\(\approx\) |
\(0.4687378232\) |
| \(L(1)\) |
\(\approx\) |
\(0.4687378232\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1123 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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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
−20.90550047465904716000457860402, −20.49713684300492621328996769773, −19.287752058938306333266881956232, −18.7750310993437054822209620334, −18.01230935737493917054006132221, −17.41126090858769135096746406355, −16.51345629634057498125491948447, −16.056571530689430943890762078, −15.12235746741460202288612747209, −14.56660820383625519034822458601, −12.884328181810085682914348557624, −12.16848210574524695723732459768, −11.49444693008538735523106041279, −10.93406084279272286810475244302, −10.20626237833711940122978834144, −9.256444241024428477826077913209, −8.09845590922483934641342647116, −7.41507158458836060255228496892, −7.12947190762985980940163392316, −5.44908903746505510609678960579, −5.18782100023510762936881203609, −3.82965564879593874928673646418, −2.61790654589270747736274853646, −1.36925151115058644872558966349, −0.4790656034654731241646256300,
0.4790656034654731241646256300, 1.36925151115058644872558966349, 2.61790654589270747736274853646, 3.82965564879593874928673646418, 5.18782100023510762936881203609, 5.44908903746505510609678960579, 7.12947190762985980940163392316, 7.41507158458836060255228496892, 8.09845590922483934641342647116, 9.256444241024428477826077913209, 10.20626237833711940122978834144, 10.93406084279272286810475244302, 11.49444693008538735523106041279, 12.16848210574524695723732459768, 12.884328181810085682914348557624, 14.56660820383625519034822458601, 15.12235746741460202288612747209, 16.056571530689430943890762078, 16.51345629634057498125491948447, 17.41126090858769135096746406355, 18.01230935737493917054006132221, 18.7750310993437054822209620334, 19.287752058938306333266881956232, 20.49713684300492621328996769773, 20.90550047465904716000457860402
