| L(s) = 1 | + 0.695·3-s − 1.38·5-s − 2.51·9-s + 0.420·11-s + 1.63·13-s − 0.962·15-s − 4.89·17-s − 3.98·19-s + 6.81·23-s − 3.08·25-s − 3.83·27-s + 0.394·29-s − 5.18·31-s + 0.292·33-s + 3.85·37-s + 1.13·39-s + 41-s + 4.97·43-s + 3.47·45-s + 3.46·47-s − 3.40·51-s + 1.26·53-s − 0.581·55-s − 2.77·57-s − 14.8·59-s + 9.77·61-s − 2.25·65-s + ⋯ |
| L(s) = 1 | + 0.401·3-s − 0.618·5-s − 0.838·9-s + 0.126·11-s + 0.452·13-s − 0.248·15-s − 1.18·17-s − 0.914·19-s + 1.42·23-s − 0.617·25-s − 0.738·27-s + 0.0731·29-s − 0.931·31-s + 0.0509·33-s + 0.633·37-s + 0.181·39-s + 0.156·41-s + 0.758·43-s + 0.518·45-s + 0.504·47-s − 0.477·51-s + 0.173·53-s − 0.0784·55-s − 0.367·57-s − 1.92·59-s + 1.25·61-s − 0.279·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.397927743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.397927743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 0.695T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 11 | \( 1 - 0.420T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 43 | \( 1 - 4.97T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 - 9.77T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 2.14T + 73T^{2} \) |
| 79 | \( 1 - 7.49T + 79T^{2} \) |
| 83 | \( 1 - 4.29T + 83T^{2} \) |
| 89 | \( 1 - 3.54T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.86938204161588448521565206595, −7.24827787023271943160762322408, −6.44178532736191856356923799252, −5.85718485731088548875432205041, −4.93131306736525231150623332798, −4.17911194977690509548754153964, −3.53707590491381682380561435559, −2.69875090014518483620881730162, −1.92895471931590294166491347529, −0.55103141646752618222793139252,
0.55103141646752618222793139252, 1.92895471931590294166491347529, 2.69875090014518483620881730162, 3.53707590491381682380561435559, 4.17911194977690509548754153964, 4.93131306736525231150623332798, 5.85718485731088548875432205041, 6.44178532736191856356923799252, 7.24827787023271943160762322408, 7.86938204161588448521565206595
