| L(s) = 1 | − 2-s + 4-s + 1.41·5-s − 4.24·7-s − 8-s − 1.41·10-s + 2.82·11-s + 2·13-s + 4.24·14-s + 16-s − 2·19-s + 1.41·20-s − 2.82·22-s + 7.07·23-s − 2.99·25-s − 2·26-s − 4.24·28-s − 7.07·29-s − 4.24·31-s − 32-s − 6·35-s + 4.24·37-s + 2·38-s − 1.41·40-s − 5.65·41-s + 4·43-s + 2.82·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.632·5-s − 1.60·7-s − 0.353·8-s − 0.447·10-s + 0.852·11-s + 0.554·13-s + 1.13·14-s + 0.250·16-s − 0.458·19-s + 0.316·20-s − 0.603·22-s + 1.47·23-s − 0.599·25-s − 0.392·26-s − 0.801·28-s − 1.31·29-s − 0.762·31-s − 0.176·32-s − 1.01·35-s + 0.697·37-s + 0.324·38-s − 0.223·40-s − 0.883·41-s + 0.609·43-s + 0.426·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
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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
−7.85053664353157576852723658443, −7.03305142136945689827890039331, −6.39456659723778672080194374139, −6.07250637441345458981486098840, −5.09566465758074781924071612193, −3.77952117497332696302969086236, −3.29914723419424734847147227171, −2.25025282138793092745830315342, −1.27725230475801522652905440572, 0,
1.27725230475801522652905440572, 2.25025282138793092745830315342, 3.29914723419424734847147227171, 3.77952117497332696302969086236, 5.09566465758074781924071612193, 6.07250637441345458981486098840, 6.39456659723778672080194374139, 7.03305142136945689827890039331, 7.85053664353157576852723658443
