| L(s) = 1 | − 2.59·3-s + 2.14·5-s + 3.74·9-s − 11-s − 0.0931·13-s − 5.58·15-s + 6.05·17-s − 1.35·19-s − 3.66·23-s − 0.382·25-s − 1.93·27-s + 0.458·29-s − 0.237·31-s + 2.59·33-s + 9.57·37-s + 0.242·39-s + 5.47·41-s + 2.67·43-s + 8.05·45-s − 2.05·47-s − 15.7·51-s − 13.8·53-s − 2.14·55-s + 3.53·57-s + 14.5·59-s − 4.02·61-s − 0.200·65-s + ⋯ |
| L(s) = 1 | − 1.49·3-s + 0.961·5-s + 1.24·9-s − 0.301·11-s − 0.0258·13-s − 1.44·15-s + 1.46·17-s − 0.311·19-s − 0.763·23-s − 0.0764·25-s − 0.373·27-s + 0.0851·29-s − 0.0427·31-s + 0.452·33-s + 1.57·37-s + 0.0387·39-s + 0.855·41-s + 0.407·43-s + 1.20·45-s − 0.299·47-s − 2.20·51-s − 1.90·53-s − 0.289·55-s + 0.467·57-s + 1.89·59-s − 0.515·61-s − 0.0248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.253681617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253681617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 - 2.14T + 5T^{2} \) |
| 13 | \( 1 + 0.0931T + 13T^{2} \) |
| 17 | \( 1 - 6.05T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 + 3.66T + 23T^{2} \) |
| 29 | \( 1 - 0.458T + 29T^{2} \) |
| 31 | \( 1 + 0.237T + 31T^{2} \) |
| 37 | \( 1 - 9.57T + 37T^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 - 2.67T + 43T^{2} \) |
| 47 | \( 1 + 2.05T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 + 9.00T + 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 - 8.01T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 9.85T + 83T^{2} \) |
| 89 | \( 1 - 2.56T + 89T^{2} \) |
| 97 | \( 1 + 0.990T + 97T^{2} \) |
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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
−8.196227192601650652060962017432, −7.60581521047851832884582429217, −6.60675920963645637399335021282, −5.99830124765495045697251435664, −5.62679300232682764991169571893, −4.90929158561303792490469929355, −4.03908525153972360203997595310, −2.81642011076678271727176929300, −1.72096052814671443818138320292, −0.70003047253564213720111136364,
0.70003047253564213720111136364, 1.72096052814671443818138320292, 2.81642011076678271727176929300, 4.03908525153972360203997595310, 4.90929158561303792490469929355, 5.62679300232682764991169571893, 5.99830124765495045697251435664, 6.60675920963645637399335021282, 7.60581521047851832884582429217, 8.196227192601650652060962017432
