| L(s) = 1 | + 1.04·5-s + 7-s − 3.26·11-s + 13-s + 7.95·17-s + 3.04·19-s + 6.31·23-s − 3.90·25-s + 4.46·29-s + 3.19·31-s + 1.04·35-s − 1.26·37-s − 10.7·41-s + 6.90·43-s − 3.19·47-s + 49-s − 8.46·53-s − 3.41·55-s + 8.62·59-s − 9.31·61-s + 1.04·65-s − 0.146·67-s + 16.4·71-s − 1.63·73-s − 3.26·77-s − 14.7·79-s − 1.58·83-s + ⋯ |
| L(s) = 1 | + 0.467·5-s + 0.377·7-s − 0.985·11-s + 0.277·13-s + 1.92·17-s + 0.698·19-s + 1.31·23-s − 0.781·25-s + 0.828·29-s + 0.573·31-s + 0.176·35-s − 0.208·37-s − 1.68·41-s + 1.05·43-s − 0.465·47-s + 0.142·49-s − 1.16·53-s − 0.460·55-s + 1.12·59-s − 1.19·61-s + 0.129·65-s − 0.0179·67-s + 1.95·71-s − 0.191·73-s − 0.372·77-s − 1.66·79-s − 0.173·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.506270551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506270551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.04T + 5T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 1.26T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 6.90T + 43T^{2} \) |
| 47 | \( 1 + 3.19T + 47T^{2} \) |
| 53 | \( 1 + 8.46T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 0.146T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 1.63T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 1.58T + 83T^{2} \) |
| 89 | \( 1 + 2.66T + 89T^{2} \) |
| 97 | \( 1 + 1.34T + 97T^{2} \) |
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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
−7.990288539915073606735240212843, −7.41898982503244765187074639358, −6.61104473782239021376095029053, −5.66051773139312022236227176632, −5.31256781033826296966953699174, −4.56843735438606604467318465215, −3.36389886891402316830682116940, −2.89419216713637773271397663655, −1.74097654536974917123883100715, −0.859246263859177735863788214517,
0.859246263859177735863788214517, 1.74097654536974917123883100715, 2.89419216713637773271397663655, 3.36389886891402316830682116940, 4.56843735438606604467318465215, 5.31256781033826296966953699174, 5.66051773139312022236227176632, 6.61104473782239021376095029053, 7.41898982503244765187074639358, 7.990288539915073606735240212843
