| L(s) = 1 | − 2-s − 2.44·3-s + 4-s + 3.01·5-s + 2.44·6-s − 8-s + 2.96·9-s − 3.01·10-s + 2.12·11-s − 2.44·12-s + 1.81·13-s − 7.36·15-s + 16-s − 1.05·17-s − 2.96·18-s + 3.34·19-s + 3.01·20-s − 2.12·22-s + 6.27·23-s + 2.44·24-s + 4.09·25-s − 1.81·26-s + 0.0948·27-s − 3.63·29-s + 7.36·30-s − 0.813·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.40·3-s + 0.5·4-s + 1.34·5-s + 0.996·6-s − 0.353·8-s + 0.987·9-s − 0.953·10-s + 0.640·11-s − 0.704·12-s + 0.504·13-s − 1.90·15-s + 0.250·16-s − 0.255·17-s − 0.697·18-s + 0.768·19-s + 0.674·20-s − 0.452·22-s + 1.30·23-s + 0.498·24-s + 0.818·25-s − 0.356·26-s + 0.0182·27-s − 0.675·29-s + 1.34·30-s − 0.146·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.203832745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203832745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 3.01T + 5T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 1.81T + 13T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 - 6.27T + 23T^{2} \) |
| 29 | \( 1 + 3.63T + 29T^{2} \) |
| 31 | \( 1 + 0.813T + 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 43 | \( 1 - 9.64T + 43T^{2} \) |
| 47 | \( 1 + 6.55T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 6.34T + 67T^{2} \) |
| 71 | \( 1 - 9.85T + 71T^{2} \) |
| 73 | \( 1 + 0.436T + 73T^{2} \) |
| 79 | \( 1 - 4.59T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 + 9.20T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.711444828881270331537637769255, −7.53930885288640990643419748344, −6.72470896226477813849561382928, −6.33848346064403343769220596072, −5.50976987542722673450074506271, −5.18099626369927360156514022681, −3.90908145947809702930143165337, −2.66180041728249802725321661806, −1.56239427802695792026055754221, −0.814914775493938359509245116884,
0.814914775493938359509245116884, 1.56239427802695792026055754221, 2.66180041728249802725321661806, 3.90908145947809702930143165337, 5.18099626369927360156514022681, 5.50976987542722673450074506271, 6.33848346064403343769220596072, 6.72470896226477813849561382928, 7.53930885288640990643419748344, 8.711444828881270331537637769255
