| L(s) = 1 | + 3-s + 9-s − 11-s − 2·17-s + 6·19-s − 8·23-s + 27-s − 4·31-s − 33-s + 2·37-s − 8·41-s − 8·43-s − 12·47-s − 2·51-s + 10·53-s + 6·57-s + 12·59-s + 2·61-s + 4·67-s − 8·69-s − 8·71-s − 12·73-s − 10·79-s + 81-s − 2·83-s + 6·89-s − 4·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.485·17-s + 1.37·19-s − 1.66·23-s + 0.192·27-s − 0.718·31-s − 0.174·33-s + 0.328·37-s − 1.24·41-s − 1.21·43-s − 1.75·47-s − 0.280·51-s + 1.37·53-s + 0.794·57-s + 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.963·69-s − 0.949·71-s − 1.40·73-s − 1.12·79-s + 1/9·81-s − 0.219·83-s + 0.635·89-s − 0.414·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.632080507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632080507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.27697676964917, −13.04316121936313, −12.23589723636217, −11.80802112912974, −11.52514524978890, −10.90120305564621, −10.18959342257210, −9.861808575895310, −9.656078162925135, −8.769523840478751, −8.485857676268132, −8.040386919214457, −7.430425380231806, −7.045857483698747, −6.493249370247817, −5.849846156365137, −5.279869635896156, −4.914372237045790, −4.005766047740145, −3.813084448403452, −3.031609721552702, −2.605121518527827, −1.793365690959424, −1.434522172593317, −0.3412403258450503,
0.3412403258450503, 1.434522172593317, 1.793365690959424, 2.605121518527827, 3.031609721552702, 3.813084448403452, 4.005766047740145, 4.914372237045790, 5.279869635896156, 5.849846156365137, 6.493249370247817, 7.045857483698747, 7.430425380231806, 8.040386919214457, 8.485857676268132, 8.769523840478751, 9.656078162925135, 9.861808575895310, 10.18959342257210, 10.90120305564621, 11.52514524978890, 11.80802112912974, 12.23589723636217, 13.04316121936313, 13.27697676964917
