L(s) = 1 | + 2-s + 2.14·3-s + 4-s + 5-s + 2.14·6-s − 4.86·7-s + 8-s + 1.60·9-s + 10-s − 11-s + 2.14·12-s + 0.772·13-s − 4.86·14-s + 2.14·15-s + 16-s + 3.64·17-s + 1.60·18-s + 6.17·19-s + 20-s − 10.4·21-s − 22-s + 1.22·23-s + 2.14·24-s + 25-s + 0.772·26-s − 2.99·27-s − 4.86·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.23·3-s + 0.5·4-s + 0.447·5-s + 0.875·6-s − 1.83·7-s + 0.353·8-s + 0.534·9-s + 0.316·10-s − 0.301·11-s + 0.619·12-s + 0.214·13-s − 1.30·14-s + 0.553·15-s + 0.250·16-s + 0.884·17-s + 0.377·18-s + 1.41·19-s + 0.223·20-s − 2.27·21-s − 0.213·22-s + 0.255·23-s + 0.437·24-s + 0.200·25-s + 0.151·26-s − 0.576·27-s − 0.919·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.564925339\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.564925339\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 2.14T + 3T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 13 | \( 1 - 0.772T + 13T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 19 | \( 1 - 6.17T + 19T^{2} \) |
| 23 | \( 1 - 1.22T + 23T^{2} \) |
| 29 | \( 1 - 8.75T + 29T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 47 | \( 1 + 0.956T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 1.28T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 2.42T + 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 + 3.19T + 89T^{2} \) |
| 97 | \( 1 + 9.25T + 97T^{2} \) |
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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
−8.242844369647639682509710433164, −7.50199915953742004725418158140, −6.85531699248844793936845521427, −6.01100406700725422726084026576, −5.53738619901225416889952703516, −4.35887706200938652015084585398, −3.41877117434027007029548890128, −3.00917912685283244856454740530, −2.48481097891978531670640045817, −1.02375406043033238903659347373,
1.02375406043033238903659347373, 2.48481097891978531670640045817, 3.00917912685283244856454740530, 3.41877117434027007029548890128, 4.35887706200938652015084585398, 5.53738619901225416889952703516, 6.01100406700725422726084026576, 6.85531699248844793936845521427, 7.50199915953742004725418158140, 8.242844369647639682509710433164