| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 13-s − 15-s + 17-s + 6·19-s + 2·21-s + 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s − 2·35-s + 6·37-s − 39-s − 6·41-s + 2·43-s + 45-s − 8·47-s − 3·49-s − 51-s − 2·53-s − 6·57-s + 10·59-s + 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 1.37·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.304·43-s + 0.149·45-s − 1.16·47-s − 3/7·49-s − 0.140·51-s − 0.274·53-s − 0.794·57-s + 1.30·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.24552057425744, −12.83403031972328, −12.34728846166409, −11.78045315753743, −11.38759853889816, −10.98191721194956, −10.38401714324935, −9.857229768569076, −9.655148118811289, −9.157299191927754, −8.524859982750110, −8.047186460828086, −7.387675840362135, −6.847424783970421, −6.642815868242780, −5.948215767058835, −5.512613371806817, −5.090250877537268, −4.594199063816060, −3.716456545130090, −3.398808335997061, −2.789387040971975, −2.136149375421393, −1.281707970266059, −0.8879093412131098, 0,
0.8879093412131098, 1.281707970266059, 2.136149375421393, 2.789387040971975, 3.398808335997061, 3.716456545130090, 4.594199063816060, 5.090250877537268, 5.512613371806817, 5.948215767058835, 6.642815868242780, 6.847424783970421, 7.387675840362135, 8.047186460828086, 8.524859982750110, 9.157299191927754, 9.655148118811289, 9.857229768569076, 10.38401714324935, 10.98191721194956, 11.38759853889816, 11.78045315753743, 12.34728846166409, 12.83403031972328, 13.24552057425744
