L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s − 4·11-s + 6·13-s − 2·15-s + 2·17-s + 8·19-s − 4·21-s + 6·23-s + 25-s + 4·27-s + 2·29-s − 4·31-s + 8·33-s + 2·35-s − 2·37-s − 12·39-s − 10·41-s − 2·43-s + 45-s + 2·47-s − 3·49-s − 4·51-s − 2·53-s − 4·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.516·15-s + 0.485·17-s + 1.83·19-s − 0.872·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.718·31-s + 1.39·33-s + 0.338·35-s − 0.328·37-s − 1.92·39-s − 1.56·41-s − 0.304·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.064744581\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.064744581\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−11.37068317400632977991374841521, −10.95278774134901337773577917803, −10.05870679171919717055814686754, −8.781404856404838385729722537924, −7.78598962640623119361177800709, −6.59250531831024987949367010419, −5.42261684437556226200264710088, −5.12441036610944036341041015255, −3.23319368204924250795089625252, −1.23403073899346067943514724338,
1.23403073899346067943514724338, 3.23319368204924250795089625252, 5.12441036610944036341041015255, 5.42261684437556226200264710088, 6.59250531831024987949367010419, 7.78598962640623119361177800709, 8.781404856404838385729722537924, 10.05870679171919717055814686754, 10.95278774134901337773577917803, 11.37068317400632977991374841521