L(s) = 1 | + 2-s − 2.24·3-s + 4-s + 5-s − 2.24·6-s + 7-s + 8-s + 2.05·9-s + 10-s − 2.24·12-s + 14-s − 2.24·15-s + 16-s + 1.05·17-s + 2.05·18-s + 5.19·19-s + 20-s − 2.24·21-s + 5.30·23-s − 2.24·24-s + 25-s + 2.11·27-s + 28-s − 3.30·29-s − 2.24·30-s − 4.61·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.447·5-s − 0.918·6-s + 0.377·7-s + 0.353·8-s + 0.686·9-s + 0.316·10-s − 0.649·12-s + 0.267·14-s − 0.580·15-s + 0.250·16-s + 0.256·17-s + 0.485·18-s + 1.19·19-s + 0.223·20-s − 0.490·21-s + 1.10·23-s − 0.459·24-s + 0.200·25-s + 0.407·27-s + 0.188·28-s − 0.614·29-s − 0.410·30-s − 0.828·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.502671552\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.502671552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 - 5.30T + 23T^{2} \) |
| 29 | \( 1 + 3.30T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 - 5.30T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 + 5.55T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 + 7.92T + 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 - 5.43T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.55T + 73T^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 - 1.88T + 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−7.49126556573530054347421464970, −6.92448700348144580472243040666, −6.18074621043972761885763368861, −5.61620848537844090051851065473, −5.12721651179225672728411577685, −4.59247218907234723711885998001, −3.56848308397155174245207393679, −2.75856274864874794228793228801, −1.64528506913175141783428740936, −0.78009882070989663670040713975,
0.78009882070989663670040713975, 1.64528506913175141783428740936, 2.75856274864874794228793228801, 3.56848308397155174245207393679, 4.59247218907234723711885998001, 5.12721651179225672728411577685, 5.61620848537844090051851065473, 6.18074621043972761885763368861, 6.92448700348144580472243040666, 7.49126556573530054347421464970