| L(s) = 1 | − 3-s + 5-s − 2·9-s + 2·11-s − 5·13-s − 15-s − 4·17-s − 2·19-s − 23-s + 25-s + 5·27-s − 3·29-s + 7·31-s − 2·33-s − 2·37-s + 5·39-s − 9·41-s − 4·43-s − 2·45-s − 9·47-s − 7·49-s + 4·51-s − 6·53-s + 2·55-s + 2·57-s + 2·61-s − 5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.603·11-s − 1.38·13-s − 0.258·15-s − 0.970·17-s − 0.458·19-s − 0.208·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 1.25·31-s − 0.348·33-s − 0.328·37-s + 0.800·39-s − 1.40·41-s − 0.609·43-s − 0.298·45-s − 1.31·47-s − 49-s + 0.560·51-s − 0.824·53-s + 0.269·55-s + 0.264·57-s + 0.256·61-s − 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−9.751717548264165206005689480655, −8.888571682998555727592360420835, −8.053192018389666773511374275467, −6.80103166369594138661595370832, −6.32853967519614382571520985105, −5.21695658707035907601168010128, −4.54380651182526809306398081482, −3.07140927984180806909538582914, −1.90962333363244051621533466532, 0,
1.90962333363244051621533466532, 3.07140927984180806909538582914, 4.54380651182526809306398081482, 5.21695658707035907601168010128, 6.32853967519614382571520985105, 6.80103166369594138661595370832, 8.053192018389666773511374275467, 8.888571682998555727592360420835, 9.751717548264165206005689480655
