| L(s) = 1 | − 1.22·3-s − 0.177·5-s − 3.50·7-s − 1.49·9-s + 2.72·11-s + 5.04·13-s + 0.217·15-s − 7.46·17-s − 8.22·19-s + 4.29·21-s + 4.99·23-s − 4.96·25-s + 5.51·27-s − 8.04·29-s + 6.68·31-s − 3.34·33-s + 0.623·35-s − 10.9·37-s − 6.17·39-s − 1.44·41-s + 8.34·43-s + 0.266·45-s − 47-s + 5.29·49-s + 9.14·51-s − 0.281·53-s − 0.485·55-s + ⋯ |
| L(s) = 1 | − 0.707·3-s − 0.0794·5-s − 1.32·7-s − 0.499·9-s + 0.822·11-s + 1.39·13-s + 0.0562·15-s − 1.80·17-s − 1.88·19-s + 0.937·21-s + 1.04·23-s − 0.993·25-s + 1.06·27-s − 1.49·29-s + 1.20·31-s − 0.582·33-s + 0.105·35-s − 1.79·37-s − 0.989·39-s − 0.225·41-s + 1.27·43-s + 0.0396·45-s − 0.145·47-s + 0.756·49-s + 1.28·51-s − 0.0386·53-s − 0.0654·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5883705158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5883705158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 + 0.177T + 5T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 + 8.22T + 19T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 + 8.04T + 29T^{2} \) |
| 31 | \( 1 - 6.68T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 1.44T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 53 | \( 1 + 0.281T + 53T^{2} \) |
| 59 | \( 1 - 7.20T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 - 4.06T + 67T^{2} \) |
| 71 | \( 1 + 2.32T + 71T^{2} \) |
| 73 | \( 1 - 3.81T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.287036842896744269623993078552, −6.97015155749275099446081799796, −6.48706941175337458601528284440, −6.20435088334181361100888953366, −5.40394371943761451080214297380, −4.24203209812589209150683640876, −3.82729303131138082549799275365, −2.83946232729938257513381956296, −1.80602102704253740333801323743, −0.40251168137403020153814848186,
0.40251168137403020153814848186, 1.80602102704253740333801323743, 2.83946232729938257513381956296, 3.82729303131138082549799275365, 4.24203209812589209150683640876, 5.40394371943761451080214297380, 6.20435088334181361100888953366, 6.48706941175337458601528284440, 6.97015155749275099446081799796, 8.287036842896744269623993078552
