| L(s) = 1 | + 2.84·3-s + 5-s + 4.84·7-s + 5.11·9-s − 2.34·11-s − 4.46·13-s + 2.84·15-s − 0.461·17-s + 2.50·19-s + 13.8·21-s − 7.69·23-s + 25-s + 6.04·27-s + 7.69·29-s + 9.43·31-s − 6.67·33-s + 4.84·35-s + 37-s − 12.7·39-s + 8.34·41-s − 11.9·43-s + 5.11·45-s − 0.0742·47-s + 16.5·49-s − 1.31·51-s + 5.35·53-s − 2.34·55-s + ⋯ |
| L(s) = 1 | + 1.64·3-s + 0.447·5-s + 1.83·7-s + 1.70·9-s − 0.706·11-s − 1.23·13-s + 0.735·15-s − 0.112·17-s + 0.575·19-s + 3.01·21-s − 1.60·23-s + 0.200·25-s + 1.16·27-s + 1.42·29-s + 1.69·31-s − 1.16·33-s + 0.819·35-s + 0.164·37-s − 2.03·39-s + 1.30·41-s − 1.82·43-s + 0.763·45-s − 0.0108·47-s + 2.35·49-s − 0.184·51-s + 0.735·53-s − 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.309137056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.309137056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2.84T + 3T^{2} \) |
| 7 | \( 1 - 4.84T + 7T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 + 0.461T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 23 | \( 1 + 7.69T + 23T^{2} \) |
| 29 | \( 1 - 7.69T + 29T^{2} \) |
| 31 | \( 1 - 9.43T + 31T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 0.0742T + 47T^{2} \) |
| 53 | \( 1 - 5.35T + 53T^{2} \) |
| 59 | \( 1 - 3.73T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 + 4.04T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 8.50T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 7.93T + 97T^{2} \) |
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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
−8.326721108522978872455808952508, −8.221438488951092764948968895073, −7.63535454152769641647475029607, −6.74409376871881989087091647568, −5.42291545901554954663548117262, −4.74339821399949667739476278709, −4.06163134422965226458281426861, −2.66153156867704041960025321639, −2.36126271062246058171525404405, −1.34393085003879774748073575820,
1.34393085003879774748073575820, 2.36126271062246058171525404405, 2.66153156867704041960025321639, 4.06163134422965226458281426861, 4.74339821399949667739476278709, 5.42291545901554954663548117262, 6.74409376871881989087091647568, 7.63535454152769641647475029607, 8.221438488951092764948968895073, 8.326721108522978872455808952508
