| L(s) = 1 | − 0.698·3-s + 3.66·5-s + 7-s − 2.51·9-s − 3.03·11-s − 2.40·13-s − 2.56·15-s + 17-s − 1.71·19-s − 0.698·21-s − 6.60·23-s + 8.43·25-s + 3.85·27-s + 5.84·29-s + 0.381·31-s + 2.12·33-s + 3.66·35-s − 6.48·37-s + 1.68·39-s − 5.75·41-s − 6.74·43-s − 9.20·45-s − 5.73·47-s + 49-s − 0.698·51-s − 4.98·53-s − 11.1·55-s + ⋯ |
| L(s) = 1 | − 0.403·3-s + 1.63·5-s + 0.377·7-s − 0.837·9-s − 0.915·11-s − 0.667·13-s − 0.661·15-s + 0.242·17-s − 0.394·19-s − 0.152·21-s − 1.37·23-s + 1.68·25-s + 0.741·27-s + 1.08·29-s + 0.0684·31-s + 0.369·33-s + 0.619·35-s − 1.06·37-s + 0.269·39-s − 0.898·41-s − 1.02·43-s − 1.37·45-s − 0.836·47-s + 0.142·49-s − 0.0978·51-s − 0.684·53-s − 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 0.698T + 3T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 2.40T + 13T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 + 6.60T + 23T^{2} \) |
| 29 | \( 1 - 5.84T + 29T^{2} \) |
| 31 | \( 1 - 0.381T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 + 5.73T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 - 0.822T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 9.22T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 3.08T + 97T^{2} \) |
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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
−8.352683610766871906180094950255, −7.31770188514030776751897753197, −6.39404586226741169894251711592, −5.88099096647492362373708314054, −5.19974377598806000704353081005, −4.69764549088495935811265258980, −3.16959992655758524457932838461, −2.37987763838462102201537354296, −1.62196117090783647827001481208, 0,
1.62196117090783647827001481208, 2.37987763838462102201537354296, 3.16959992655758524457932838461, 4.69764549088495935811265258980, 5.19974377598806000704353081005, 5.88099096647492362373708314054, 6.39404586226741169894251711592, 7.31770188514030776751897753197, 8.352683610766871906180094950255
