| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2·13-s − 14-s + 16-s − 6·17-s − 19-s − 6·23-s + 2·26-s + 28-s − 6·29-s + 5·31-s − 32-s + 6·34-s + 7·37-s + 38-s + 12·41-s − 11·43-s + 6·46-s − 12·47-s − 6·49-s − 2·52-s − 56-s + 6·58-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.229·19-s − 1.25·23-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 0.898·31-s − 0.176·32-s + 1.02·34-s + 1.15·37-s + 0.162·38-s + 1.87·41-s − 1.67·43-s + 0.884·46-s − 1.75·47-s − 6/7·49-s − 0.277·52-s − 0.133·56-s + 0.787·58-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−9.365023266490086554487001161364, −8.249024550459436728493681503388, −7.86351767062541232131960218076, −6.77240350899289718285628928770, −6.13144008193075785449250458150, −4.93922000774390898350676467392, −4.05329998572738233001654346526, −2.65515308426446773842297813381, −1.72742911706225949144999864292, 0,
1.72742911706225949144999864292, 2.65515308426446773842297813381, 4.05329998572738233001654346526, 4.93922000774390898350676467392, 6.13144008193075785449250458150, 6.77240350899289718285628928770, 7.86351767062541232131960218076, 8.249024550459436728493681503388, 9.365023266490086554487001161364
