| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 5·11-s − 13-s − 15-s − 3·19-s − 21-s − 23-s − 4·25-s + 5·27-s − 10·29-s + 4·31-s + 5·33-s + 35-s + 6·37-s + 39-s − 2·43-s − 2·45-s − 6·47-s + 49-s + 12·53-s − 5·55-s + 3·57-s − 6·59-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s − 0.688·19-s − 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.962·27-s − 1.85·29-s + 0.718·31-s + 0.870·33-s + 0.169·35-s + 0.986·37-s + 0.160·39-s − 0.304·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s + 1.64·53-s − 0.674·55-s + 0.397·57-s − 0.781·59-s − 0.251·63-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 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.51767895947789, −13.24296678152557, −12.81629010184421, −12.22082663035225, −11.69528003089523, −11.24746461291917, −10.90899582776399, −10.28211510620032, −10.02927083286091, −9.347833176622969, −8.867599597790845, −8.109104180747597, −7.966584610279597, −7.360616542793982, −6.684678810318189, −6.045427897217565, −5.749066988360018, −5.189211039599451, −4.861940874993625, −4.113716430204857, −3.473092269400504, −2.638571389024318, −2.310410977678216, −1.677962964673616, −0.6296099474105745, 0,
0.6296099474105745, 1.677962964673616, 2.310410977678216, 2.638571389024318, 3.473092269400504, 4.113716430204857, 4.861940874993625, 5.189211039599451, 5.749066988360018, 6.045427897217565, 6.684678810318189, 7.360616542793982, 7.966584610279597, 8.109104180747597, 8.867599597790845, 9.347833176622969, 10.02927083286091, 10.28211510620032, 10.90899582776399, 11.24746461291917, 11.69528003089523, 12.22082663035225, 12.81629010184421, 13.24296678152557, 13.51767895947789
