| L(s) = 1 | + 2-s + 4-s + 5-s + 3.21·7-s + 8-s + 10-s + 6.35·11-s + 6.24·13-s + 3.21·14-s + 16-s − 1.21·17-s − 6.95·19-s + 20-s + 6.35·22-s − 4.24·23-s + 25-s + 6.24·26-s + 3.21·28-s + 3.03·29-s + 4.59·31-s + 32-s − 1.21·34-s + 3.21·35-s + 37-s − 6.95·38-s + 40-s − 3.47·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.21·7-s + 0.353·8-s + 0.316·10-s + 1.91·11-s + 1.73·13-s + 0.859·14-s + 0.250·16-s − 0.294·17-s − 1.59·19-s + 0.223·20-s + 1.35·22-s − 0.885·23-s + 0.200·25-s + 1.22·26-s + 0.607·28-s + 0.563·29-s + 0.824·31-s + 0.176·32-s − 0.208·34-s + 0.543·35-s + 0.164·37-s − 1.12·38-s + 0.158·40-s − 0.542·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.439170805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.439170805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - 3.21T + 7T^{2} \) |
| 11 | \( 1 - 6.35T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 + 6.95T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 3.03T + 29T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 + 1.47T + 43T^{2} \) |
| 47 | \( 1 + 8.01T + 47T^{2} \) |
| 53 | \( 1 + 0.773T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 6.28T + 67T^{2} \) |
| 71 | \( 1 - 0.882T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 2.88T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 6.61T + 89T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
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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
−8.508159350949327073085208796442, −8.078538804064700503899014712302, −6.71041969765214536533343283572, −6.37526915195692524940469190120, −5.73429970692393306120954643605, −4.40210054818457493706053114869, −4.30808840965176781475774275941, −3.21082279885002450848933891192, −1.80431656127142705863334910735, −1.39701271638754394976802518131,
1.39701271638754394976802518131, 1.80431656127142705863334910735, 3.21082279885002450848933891192, 4.30808840965176781475774275941, 4.40210054818457493706053114869, 5.73429970692393306120954643605, 6.37526915195692524940469190120, 6.71041969765214536533343283572, 8.078538804064700503899014712302, 8.508159350949327073085208796442
