| L(s) = 1 | + 2-s + 1.72·3-s + 4-s − 0.551·5-s + 1.72·6-s + 1.69·7-s + 8-s − 0.0250·9-s − 0.551·10-s − 5.19·11-s + 1.72·12-s − 0.613·13-s + 1.69·14-s − 0.950·15-s + 16-s − 3.04·17-s − 0.0250·18-s − 0.467·19-s − 0.551·20-s + 2.91·21-s − 5.19·22-s − 4.06·23-s + 1.72·24-s − 4.69·25-s − 0.613·26-s − 5.21·27-s + 1.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.995·3-s + 0.5·4-s − 0.246·5-s + 0.704·6-s + 0.639·7-s + 0.353·8-s − 0.00836·9-s − 0.174·10-s − 1.56·11-s + 0.497·12-s − 0.170·13-s + 0.452·14-s − 0.245·15-s + 0.250·16-s − 0.737·17-s − 0.00591·18-s − 0.107·19-s − 0.123·20-s + 0.636·21-s − 1.10·22-s − 0.847·23-s + 0.352·24-s − 0.939·25-s − 0.120·26-s − 1.00·27-s + 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 + 0.551T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 0.613T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + 0.467T + 19T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 - 8.54T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 2.23T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 0.373T + 43T^{2} \) |
| 47 | \( 1 + 2.22T + 47T^{2} \) |
| 53 | \( 1 + 2.15T + 53T^{2} \) |
| 59 | \( 1 - 3.29T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 - 6.20T + 67T^{2} \) |
| 71 | \( 1 - 8.57T + 71T^{2} \) |
| 73 | \( 1 - 5.59T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 5.29T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.969500399110476481644873033319, −7.14942764502456149019435644930, −6.25308000080856186569562875490, −5.34203882249374038553925422766, −4.86863999970782268081042945456, −3.93692506684595501204536411290, −3.25115084628542103407619878754, −2.37509842100980859137289420642, −1.90223717733734314565405868437, 0,
1.90223717733734314565405868437, 2.37509842100980859137289420642, 3.25115084628542103407619878754, 3.93692506684595501204536411290, 4.86863999970782268081042945456, 5.34203882249374038553925422766, 6.25308000080856186569562875490, 7.14942764502456149019435644930, 7.969500399110476481644873033319
