| L(s) = 1 | + 0.456·2-s − 3-s − 1.79·4-s − 5-s − 0.456·6-s + 7-s − 1.72·8-s + 9-s − 0.456·10-s − 4.03·11-s + 1.79·12-s − 3.30·13-s + 0.456·14-s + 15-s + 2.79·16-s + 1.85·17-s + 0.456·18-s − 4.75·19-s + 1.79·20-s − 21-s − 1.84·22-s − 23-s + 1.72·24-s + 25-s − 1.50·26-s − 27-s − 1.79·28-s + ⋯ |
| L(s) = 1 | + 0.322·2-s − 0.577·3-s − 0.895·4-s − 0.447·5-s − 0.186·6-s + 0.377·7-s − 0.611·8-s + 0.333·9-s − 0.144·10-s − 1.21·11-s + 0.517·12-s − 0.915·13-s + 0.121·14-s + 0.258·15-s + 0.698·16-s + 0.449·17-s + 0.107·18-s − 1.09·19-s + 0.400·20-s − 0.218·21-s − 0.392·22-s − 0.208·23-s + 0.353·24-s + 0.200·25-s − 0.295·26-s − 0.192·27-s − 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6965467927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6965467927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.456T + 2T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 + 4.66T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 0.630T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 6.42T + 59T^{2} \) |
| 61 | \( 1 + 1.18T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 8.99T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 9.93T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.856576138692414875421505733441, −8.119951567094704436503903082779, −7.56565770606657627984756190905, −6.53946471747950754178944084345, −5.53734282829967030334861942834, −4.99380041006094313908144217276, −4.36440290800721127327589993157, −3.39498666756076663476993396330, −2.21397479752081841353261513419, −0.50620076718392544027185222817,
0.50620076718392544027185222817, 2.21397479752081841353261513419, 3.39498666756076663476993396330, 4.36440290800721127327589993157, 4.99380041006094313908144217276, 5.53734282829967030334861942834, 6.53946471747950754178944084345, 7.56565770606657627984756190905, 8.119951567094704436503903082779, 8.856576138692414875421505733441
