| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 2·13-s + 16-s + 6·17-s + 4·19-s + 2·20-s − 8·23-s − 25-s − 2·26-s − 2·29-s − 31-s − 32-s − 6·34-s + 10·37-s − 4·38-s − 2·40-s + 6·41-s + 8·43-s + 8·46-s + 8·47-s − 7·49-s + 50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.447·20-s − 1.66·23-s − 1/5·25-s − 0.392·26-s − 0.371·29-s − 0.179·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s − 0.648·38-s − 0.316·40-s + 0.937·41-s + 1.21·43-s + 1.17·46-s + 1.16·47-s − 49-s + 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292749798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292749798\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.50978028730293557207764308625, −9.792819761543788827804434951174, −9.258085561472781634099322215629, −8.059614510201906936269350354648, −7.42375835124516750954422223560, −6.01709210768183680394437750392, −5.66012783592677814232561850819, −3.93624561426330149363253797109, −2.56538619972077555026069053800, −1.25027517360414188495745145887,
1.25027517360414188495745145887, 2.56538619972077555026069053800, 3.93624561426330149363253797109, 5.66012783592677814232561850819, 6.01709210768183680394437750392, 7.42375835124516750954422223560, 8.059614510201906936269350354648, 9.258085561472781634099322215629, 9.792819761543788827804434951174, 10.50978028730293557207764308625
