| L(s) = 1 | + 3·3-s − 7-s + 6·9-s − 6·11-s − 5·13-s + 2·17-s − 2·19-s − 3·21-s − 23-s + 9·27-s − 29-s + 9·31-s − 18·33-s − 12·37-s − 15·39-s + 41-s − 2·43-s − 7·47-s + 49-s + 6·51-s − 4·53-s − 6·57-s − 12·59-s − 8·61-s − 6·63-s + 12·67-s − 3·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s − 1.80·11-s − 1.38·13-s + 0.485·17-s − 0.458·19-s − 0.654·21-s − 0.208·23-s + 1.73·27-s − 0.185·29-s + 1.61·31-s − 3.13·33-s − 1.97·37-s − 2.40·39-s + 0.156·41-s − 0.304·43-s − 1.02·47-s + 1/7·49-s + 0.840·51-s − 0.549·53-s − 0.794·57-s − 1.56·59-s − 1.02·61-s − 0.755·63-s + 1.46·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.498328681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498328681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.86427902999738, −12.40432798084310, −12.29199988120100, −11.47647409670707, −10.68575779950029, −10.30861134989164, −10.05896100737297, −9.482360295727580, −9.229759843684742, −8.453167382304562, −8.081383339555363, −7.888366762071985, −7.343847127951557, −6.868352198676768, −6.339255010047273, −5.526221817993382, −4.923680625600408, −4.721943102042671, −3.930606560012791, −3.293870467131720, −2.954987008029904, −2.484575513726584, −2.065784262540194, −1.395041839834546, −0.2687018785153200,
0.2687018785153200, 1.395041839834546, 2.065784262540194, 2.484575513726584, 2.954987008029904, 3.293870467131720, 3.930606560012791, 4.721943102042671, 4.923680625600408, 5.526221817993382, 6.339255010047273, 6.868352198676768, 7.343847127951557, 7.888366762071985, 8.081383339555363, 8.453167382304562, 9.229759843684742, 9.482360295727580, 10.05896100737297, 10.30861134989164, 10.68575779950029, 11.47647409670707, 12.29199988120100, 12.40432798084310, 12.86427902999738
