| L(s) = 1 | + 2.80·3-s − 5-s + 7-s + 4.87·9-s − 2.18·11-s + 13-s − 2.80·15-s + 3.87·17-s − 2.44·19-s + 2.80·21-s + 5.42·23-s + 25-s + 5.25·27-s + 3.38·29-s + 0.805·31-s − 6.13·33-s − 35-s + 2.75·37-s + 2.80·39-s + 6.63·41-s + 0.574·43-s − 4.87·45-s − 7.74·47-s + 49-s + 10.8·51-s + 5.61·53-s + 2.18·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s − 0.447·5-s + 0.377·7-s + 1.62·9-s − 0.659·11-s + 0.277·13-s − 0.724·15-s + 0.939·17-s − 0.561·19-s + 0.612·21-s + 1.13·23-s + 0.200·25-s + 1.01·27-s + 0.627·29-s + 0.144·31-s − 1.06·33-s − 0.169·35-s + 0.452·37-s + 0.449·39-s + 1.03·41-s + 0.0875·43-s − 0.726·45-s − 1.12·47-s + 0.142·49-s + 1.52·51-s + 0.770·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.483841916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.483841916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 - 0.805T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 - 6.63T + 41T^{2} \) |
| 43 | \( 1 - 0.574T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 - 5.61T + 53T^{2} \) |
| 59 | \( 1 + 4.93T + 59T^{2} \) |
| 61 | \( 1 - 8.50T + 61T^{2} \) |
| 67 | \( 1 - 1.73T + 67T^{2} \) |
| 71 | \( 1 - 7.03T + 71T^{2} \) |
| 73 | \( 1 + 0.760T + 73T^{2} \) |
| 79 | \( 1 - 7.35T + 79T^{2} \) |
| 83 | \( 1 - 5.61T + 83T^{2} \) |
| 89 | \( 1 + 3.51T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.398171364800143130577849045404, −7.968225061133117528217609402753, −7.40100373926183689510569052765, −6.54236869800693177925976482673, −5.37492720412813657602275363640, −4.53466836261590589081905962201, −3.70539654674927538114023919672, −2.98163356352852521462747466773, −2.26453389926418197909948743851, −1.06837385016726601785082576264,
1.06837385016726601785082576264, 2.26453389926418197909948743851, 2.98163356352852521462747466773, 3.70539654674927538114023919672, 4.53466836261590589081905962201, 5.37492720412813657602275363640, 6.54236869800693177925976482673, 7.40100373926183689510569052765, 7.968225061133117528217609402753, 8.398171364800143130577849045404
