L(s) = 1 | + 2-s + 3.17·3-s + 4-s − 0.910·5-s + 3.17·6-s + 8-s + 7.05·9-s − 0.910·10-s − 11-s + 3.17·12-s − 2.96·13-s − 2.88·15-s + 16-s + 0.648·17-s + 7.05·18-s + 7.87·19-s − 0.910·20-s − 22-s − 2.05·23-s + 3.17·24-s − 4.17·25-s − 2.96·26-s + 12.8·27-s − 1.17·29-s − 2.88·30-s − 4.99·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.83·3-s + 0.5·4-s − 0.407·5-s + 1.29·6-s + 0.353·8-s + 2.35·9-s − 0.287·10-s − 0.301·11-s + 0.915·12-s − 0.823·13-s − 0.745·15-s + 0.250·16-s + 0.157·17-s + 1.66·18-s + 1.80·19-s − 0.203·20-s − 0.213·22-s − 0.429·23-s + 0.647·24-s − 0.834·25-s − 0.582·26-s + 2.47·27-s − 0.217·29-s − 0.527·30-s − 0.896·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.133023454\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.133023454\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 + 0.910T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 - 0.648T + 17T^{2} \) |
| 19 | \( 1 - 7.87T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 + 0.342T + 53T^{2} \) |
| 59 | \( 1 + 5.82T + 59T^{2} \) |
| 61 | \( 1 + 5.64T + 61T^{2} \) |
| 67 | \( 1 + 2.74T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 1.77T + 83T^{2} \) |
| 89 | \( 1 + 7.04T + 89T^{2} \) |
| 97 | \( 1 - 3.58T + 97T^{2} \) |
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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
−9.663877483513700942089861236876, −9.408894530494996113470875064798, −8.052349555317362432360408087283, −7.70248098819216750072784614704, −6.97169497197580808684707741616, −5.50454111774982032109522776091, −4.44968807043651748755583257321, −3.52930397207506133498347045538, −2.86350057130295272773335005231, −1.76653230745259533065121636826,
1.76653230745259533065121636826, 2.86350057130295272773335005231, 3.52930397207506133498347045538, 4.44968807043651748755583257321, 5.50454111774982032109522776091, 6.97169497197580808684707741616, 7.70248098819216750072784614704, 8.052349555317362432360408087283, 9.408894530494996113470875064798, 9.663877483513700942089861236876