| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 13-s + 15-s + 8·17-s − 5·19-s + 21-s − 4·25-s + 27-s − 5·29-s − 6·31-s + 35-s − 11·37-s − 39-s − 8·43-s + 45-s + 7·47-s + 49-s + 8·51-s + 2·53-s − 5·57-s + 7·59-s + 2·61-s + 63-s − 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 1.94·17-s − 1.14·19-s + 0.218·21-s − 4/5·25-s + 0.192·27-s − 0.928·29-s − 1.07·31-s + 0.169·35-s − 1.80·37-s − 0.160·39-s − 1.21·43-s + 0.149·45-s + 1.02·47-s + 1/7·49-s + 1.12·51-s + 0.274·53-s − 0.662·57-s + 0.911·59-s + 0.256·61-s + 0.125·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.88699690734641, −15.19034236339653, −14.60325562272202, −14.53677392222326, −13.70907336378558, −13.32312317877219, −12.66440361896692, −12.08255481542023, −11.70039398061087, −10.65953130888043, −10.45192079503806, −9.737955353646916, −9.268103769147518, −8.579053192913708, −8.079633988355017, −7.438210360662755, −6.982663364064073, −6.077052248082950, −5.458948152983378, −5.041168515524569, −3.921939535369083, −3.636423356802022, −2.687382895711676, −1.914620511185696, −1.380206938105230, 0,
1.380206938105230, 1.914620511185696, 2.687382895711676, 3.636423356802022, 3.921939535369083, 5.041168515524569, 5.458948152983378, 6.077052248082950, 6.982663364064073, 7.438210360662755, 8.079633988355017, 8.579053192913708, 9.268103769147518, 9.737955353646916, 10.45192079503806, 10.65953130888043, 11.70039398061087, 12.08255481542023, 12.66440361896692, 13.32312317877219, 13.70907336378558, 14.53677392222326, 14.60325562272202, 15.19034236339653, 15.88699690734641
