| L(s) = 1 | + 1.23·3-s − 5-s − 3.23·7-s − 1.47·9-s − 2·11-s − 4.47·13-s − 1.23·15-s + 4.47·17-s − 4.47·19-s − 4.00·21-s − 4.76·23-s + 25-s − 5.52·27-s − 2·29-s + 6.47·31-s − 2.47·33-s + 3.23·35-s + 6.94·37-s − 5.52·39-s + 12.4·41-s − 7.70·43-s + 1.47·45-s − 7.23·47-s + 3.47·49-s + 5.52·51-s + 0.472·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.713·3-s − 0.447·5-s − 1.22·7-s − 0.490·9-s − 0.603·11-s − 1.24·13-s − 0.319·15-s + 1.08·17-s − 1.02·19-s − 0.872·21-s − 0.993·23-s + 0.200·25-s − 1.06·27-s − 0.371·29-s + 1.16·31-s − 0.430·33-s + 0.546·35-s + 1.14·37-s − 0.885·39-s + 1.94·41-s − 1.17·43-s + 0.219·45-s − 1.05·47-s + 0.496·49-s + 0.774·51-s + 0.0648·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−9.837221388628321043547328959521, −9.490924190263004835834897967896, −8.185347915552936506519249314816, −7.78068794946226330057829943150, −6.59398873734543733075222380276, −5.65842614683361948484141752533, −4.34251739041583454839130835374, −3.20131978645931790964482575290, −2.47570690573753637878214592504, 0,
2.47570690573753637878214592504, 3.20131978645931790964482575290, 4.34251739041583454839130835374, 5.65842614683361948484141752533, 6.59398873734543733075222380276, 7.78068794946226330057829943150, 8.185347915552936506519249314816, 9.490924190263004835834897967896, 9.837221388628321043547328959521
