| L(s) = 1 | − 0.428·2-s − 7.81·4-s − 7·7-s + 6.77·8-s + 27.4·11-s + 46.5·13-s + 2.99·14-s + 59.6·16-s + 5.20·17-s − 91.0·19-s − 11.7·22-s − 111.·23-s − 19.9·26-s + 54.7·28-s − 0.0763·29-s + 201.·31-s − 79.7·32-s − 2.22·34-s + 312.·37-s + 38.9·38-s − 102.·41-s − 257.·43-s − 214.·44-s + 47.7·46-s − 350.·47-s + 49·49-s − 363.·52-s + ⋯ |
| L(s) = 1 | − 0.151·2-s − 0.977·4-s − 0.377·7-s + 0.299·8-s + 0.753·11-s + 0.993·13-s + 0.0572·14-s + 0.931·16-s + 0.0742·17-s − 1.09·19-s − 0.114·22-s − 1.01·23-s − 0.150·26-s + 0.369·28-s − 0.000488·29-s + 1.16·31-s − 0.440·32-s − 0.0112·34-s + 1.39·37-s + 0.166·38-s − 0.390·41-s − 0.912·43-s − 0.735·44-s + 0.153·46-s − 1.08·47-s + 0.142·49-s − 0.970·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.305901776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.305901776\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 0.428T + 8T^{2} \) |
| 11 | \( 1 - 27.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.20T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 0.0763T + 2.43e4T^{2} \) |
| 31 | \( 1 - 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 312.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 257.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 350.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 196.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 881.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 737.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 365.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 273.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 87.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 228.T + 9.12e5T^{2} \) |
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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
−8.923801977120444025581486102612, −8.478633470597195238600502389098, −7.66702812413027379716873007786, −6.39259795123317616735665721607, −6.01337204743365207448075129753, −4.70538847754145419552021846510, −4.06963301541401925601307134262, −3.21130820388420569637777411710, −1.70777748482992593916189073070, −0.58813227698998415881163990115,
0.58813227698998415881163990115, 1.70777748482992593916189073070, 3.21130820388420569637777411710, 4.06963301541401925601307134262, 4.70538847754145419552021846510, 6.01337204743365207448075129753, 6.39259795123317616735665721607, 7.66702812413027379716873007786, 8.478633470597195238600502389098, 8.923801977120444025581486102612
