| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 3·11-s + 12-s − 2·13-s + 14-s + 16-s + 2·17-s − 2·18-s − 3·19-s + 21-s + 3·22-s − 23-s + 24-s − 5·25-s − 2·26-s − 5·27-s + 28-s + 4·31-s + 32-s + 3·33-s + 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.471·18-s − 0.688·19-s + 0.218·21-s + 0.639·22-s − 0.208·23-s + 0.204·24-s − 25-s − 0.392·26-s − 0.962·27-s + 0.188·28-s + 0.718·31-s + 0.176·32-s + 0.522·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.546738139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.546738139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−12.88154531904964, −12.08753643249855, −12.03403321791489, −11.57534700152253, −11.04936956738094, −10.54527287776138, −9.997578541407475, −9.553360727633600, −8.992583569168052, −8.610500871551824, −8.057243435931700, −7.584668394179984, −7.288141151954396, −6.379800168604341, −6.191006805075322, −5.721339986735087, −4.926361521483787, −4.681205455447407, −3.999990003179831, −3.497457138897720, −3.130969655385168, −2.312712051675280, −2.044167619934868, −1.344984864435555, −0.4643775314275756,
0.4643775314275756, 1.344984864435555, 2.044167619934868, 2.312712051675280, 3.130969655385168, 3.497457138897720, 3.999990003179831, 4.681205455447407, 4.926361521483787, 5.721339986735087, 6.191006805075322, 6.379800168604341, 7.288141151954396, 7.584668394179984, 8.057243435931700, 8.610500871551824, 8.992583569168052, 9.553360727633600, 9.997578541407475, 10.54527287776138, 11.04936956738094, 11.57534700152253, 12.03403321791489, 12.08753643249855, 12.88154531904964
