| L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s − 6·13-s − 2·15-s + 2·17-s + 19-s − 4·21-s − 2·23-s + 25-s + 4·27-s + 2·29-s + 4·31-s + 2·35-s + 10·37-s + 12·39-s − 10·41-s − 6·43-s + 45-s − 6·47-s − 3·49-s − 4·51-s − 6·53-s − 2·57-s + 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s + 0.485·17-s + 0.229·19-s − 0.872·21-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.718·31-s + 0.338·35-s + 1.64·37-s + 1.92·39-s − 1.56·41-s − 0.914·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.560·51-s − 0.824·53-s − 0.264·57-s + 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.79497122909070729230975901167, −6.76092766673896308472273583644, −6.35325897786086339440052896451, −5.34581376198822801198219497435, −5.05753034422239968677813934261, −4.42370503190283266986014202128, −3.11484300587933940109704297622, −2.22121268754936937703711757367, −1.19435999771081356370334617186, 0,
1.19435999771081356370334617186, 2.22121268754936937703711757367, 3.11484300587933940109704297622, 4.42370503190283266986014202128, 5.05753034422239968677813934261, 5.34581376198822801198219497435, 6.35325897786086339440052896451, 6.76092766673896308472273583644, 7.79497122909070729230975901167
