| L(s) = 1 | − 3·3-s + 7-s + 6·9-s − 5·17-s − 7·19-s − 3·21-s − 8·23-s − 9·27-s − 3·29-s + 5·31-s + 37-s + 8·41-s − 10·43-s − 6·49-s + 15·51-s + 53-s + 21·57-s − 12·59-s − 5·61-s + 6·63-s − 4·67-s + 24·69-s + 7·71-s + 2·73-s − 4·79-s + 9·81-s + 9·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s − 1.21·17-s − 1.60·19-s − 0.654·21-s − 1.66·23-s − 1.73·27-s − 0.557·29-s + 0.898·31-s + 0.164·37-s + 1.24·41-s − 1.52·43-s − 6/7·49-s + 2.10·51-s + 0.137·53-s + 2.78·57-s − 1.56·59-s − 0.640·61-s + 0.755·63-s − 0.488·67-s + 2.88·69-s + 0.830·71-s + 0.234·73-s − 0.450·79-s + 81-s + 0.964·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.24353566618343, −14.66134492634945, −13.97498453655707, −13.34955443226030, −12.95030101792741, −12.26749579137735, −12.07602828641358, −11.31364026809753, −11.06114610092397, −10.60071906550994, −10.06240511251640, −9.528533521222006, −8.781097422966215, −8.136713608869769, −7.674457605198418, −6.808915169474078, −6.371067040894669, −6.138188682222478, −5.396330519894773, −4.784442067128031, −4.283666181631039, −3.936141247888551, −2.668603453016041, −1.915442245819545, −1.318841325038538, 0, 0,
1.318841325038538, 1.915442245819545, 2.668603453016041, 3.936141247888551, 4.283666181631039, 4.784442067128031, 5.396330519894773, 6.138188682222478, 6.371067040894669, 6.808915169474078, 7.674457605198418, 8.136713608869769, 8.781097422966215, 9.528533521222006, 10.06240511251640, 10.60071906550994, 11.06114610092397, 11.31364026809753, 12.07602828641358, 12.26749579137735, 12.95030101792741, 13.34955443226030, 13.97498453655707, 14.66134492634945, 15.24353566618343
