| L(s) = 1 | + 5·5-s + 16.5·7-s + 72.5·11-s − 59.8·13-s − 15.8·17-s − 136.·19-s − 163.·23-s + 25·25-s − 11.5·29-s − 41.0·31-s + 82.8·35-s − 242.·37-s − 57.8·41-s − 264.·43-s − 599.·47-s − 68.3·49-s − 592.·53-s + 362.·55-s − 288.·59-s + 825.·61-s − 299.·65-s + 810.·67-s + 966.·71-s + 802.·73-s + 1.20e3·77-s − 1.16e3·79-s − 502.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.894·7-s + 1.98·11-s − 1.27·13-s − 0.226·17-s − 1.64·19-s − 1.47·23-s + 0.200·25-s − 0.0742·29-s − 0.237·31-s + 0.400·35-s − 1.07·37-s − 0.220·41-s − 0.938·43-s − 1.86·47-s − 0.199·49-s − 1.53·53-s + 0.889·55-s − 0.637·59-s + 1.73·61-s − 0.571·65-s + 1.47·67-s + 1.61·71-s + 1.28·73-s + 1.77·77-s − 1.65·79-s − 0.664·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 16.5T + 343T^{2} \) |
| 11 | \( 1 - 72.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 41.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 242.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 57.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 264.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 599.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 592.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 288.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 825.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 810.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 966.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 802.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 502.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 8.76T + 7.04e5T^{2} \) |
| 97 | \( 1 - 138.T + 9.12e5T^{2} \) |
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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
−8.566132115623663905573013286983, −8.021837538953466342888944073798, −6.73819352218830634285208731099, −6.47116867401832814275804317172, −5.23072875064174911706212946819, −4.46428034023770095648626197433, −3.65930887519071794736128519526, −2.11023465298979791257950407697, −1.60699066982479982211811144054, 0,
1.60699066982479982211811144054, 2.11023465298979791257950407697, 3.65930887519071794736128519526, 4.46428034023770095648626197433, 5.23072875064174911706212946819, 6.47116867401832814275804317172, 6.73819352218830634285208731099, 8.021837538953466342888944073798, 8.566132115623663905573013286983
