| L(s) = 1 | + 7-s − 3·9-s + 13-s + 6·17-s − 4·19-s + 23-s − 5·25-s + 2·29-s + 10·31-s − 4·37-s + 10·41-s − 4·43-s − 6·47-s + 49-s + 2·53-s + 2·59-s − 10·61-s − 3·63-s − 16·67-s − 10·71-s − 2·73-s + 8·79-s + 9·81-s + 16·83-s − 8·89-s + 91-s − 4·97-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 0.277·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s − 25-s + 0.371·29-s + 1.79·31-s − 0.657·37-s + 1.56·41-s − 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s + 0.260·59-s − 1.28·61-s − 0.377·63-s − 1.95·67-s − 1.18·71-s − 0.234·73-s + 0.900·79-s + 81-s + 1.75·83-s − 0.847·89-s + 0.104·91-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.60359128377229, −13.47411384487471, −12.62020202124650, −12.11875424776487, −11.85094079230001, −11.39723845337015, −10.68645690794447, −10.46786303094973, −9.844543041648834, −9.315240141878746, −8.752579609836994, −8.291754761092504, −7.857865114029179, −7.522135369625689, −6.573328919152165, −6.278067776306810, −5.666794230060019, −5.318221271827123, −4.506353158173403, −4.192687289380097, −3.248359374975772, −3.012701658994708, −2.254248170695650, −1.545482373846534, −0.8606815290884784, 0,
0.8606815290884784, 1.545482373846534, 2.254248170695650, 3.012701658994708, 3.248359374975772, 4.192687289380097, 4.506353158173403, 5.318221271827123, 5.666794230060019, 6.278067776306810, 6.573328919152165, 7.522135369625689, 7.857865114029179, 8.291754761092504, 8.752579609836994, 9.315240141878746, 9.844543041648834, 10.46786303094973, 10.68645690794447, 11.39723845337015, 11.85094079230001, 12.11875424776487, 12.62020202124650, 13.47411384487471, 13.60359128377229
